• Each scanner component has a given rating in light-years (ly), within which radius it can detect uncloaked ships in deep space. Some scanners (known as "planet-penetrating scanners", or "penetrating scanners"/"penscans" for short) additionally have a second, lower rating, within which radius they can detect uncloaked ships in orbit of a planet, uncloaked starbases, and the statistics (owner, population, mineral concentrations, habitability values) of planets themselves. The population of Alternate Reality races' planets will always be reported as 0, no matter what it actually is. Claim Adjuster races, when viewing scan reports of a planet inhabited by a race they have set to "friend", will see the owner's habitability bars instead of their own.
  
• In addition to scanner components mounted on ships, there are a few other things that have detection capabilities:
  - Planetary scanners: inhabited planets can construct a "Planetary Scanner" from the production queue. This will use the rating(s) of the best planetary scanner component you have the tech for (it's automatically upgraded when you discover a new one).
  - AR planetary scanners: planets controlled by an Alternate Reality race cannot construct planetary scanners, but instead receive an organic scanner dependent on the population of the world. All worlds receive an ordinary scanner with a rating of (SQRT(population/10)), and worlds with an Ultra Station or Death Star starbase receive a planet-penetrating scanner with half that rating. There doesn't appear to be a cap on this formula, but very strange things happen when you put over 1 billion pop on a world.
  - SD "minefield scanning": Space Demolition races' minefields act as ordinary scanners; ships in deep space within a Space Demolition minefield will be detected by the minefield's owner as though they had been scanned.
  - PP "packet scanning": Packet Physics races' mineral packets act as scanners; packets have an ordinary and planet-penetrating rating equal to the square of the warp speed at which they are flung (e.g. a packet flung at Warp 12 will detect ships and planets within 144 ly). Packet Physics races can also detect all mineral packets and deep-space scrap in the universe.
  - IT "gate-scanning": Interstellar Traveller races' stargates act as special scanners, with a rating equal to their distance rating as a stargate. These scanners will detect planets with starbases that have stargates, and

• I've seen people arguing both that NAS is better for detecting minefields/wormholes and that penetrating scanners are better for detecting them. Given that penetrating scan range is about half of ordinary scan range, that NAS doubles ordinary scan range, and that minefields/wormholes are 75% cloaked vs. ordinary scans but uncloaked to penscans, the ultimate answer is that by and large they're about even once decent penetrating scanners appear at Elec 10. If you want a full breakdown:
  - Before penetrating scanners are available, NAS is going to have twice the detection range of non-NAS. Obviously.
  - The Ferret Scanner has a short penetrating range of 50 ly, while NAS races will have 75 ly (Possum) or 112 ly (Gazelle).
  - The Dolphin Scanner has a penetrating range of 100 ly, while NAS races will have 112 ly (Gazelle) or 137 ly (Cheetah).
  - The Elephant Scanner has a penetrating range of 200 ly, while NAS races will have 167 ly (Eagle Eye) or eventually 250 ly (Peerless).
  - Complicating this is that planetary scanners stop improving at Elec 8 for NAS races, while they continue to improve for penscan races. This means that at Elec 10, along with the Dolphin Scanner a non-NAS race is going to have the Snooper 320X planetary scanner and as such 160 ly of minefield detection from their planets. And at maxed Electronics against the NAS Peerless's 250 ly the penscan race also has the Snooper 620X and its 310 ly penscan range.
  
• Keep in mind, when putting scanners on Frigate and Galleon designs, that a second scanner adds only 19% to the first scanner's range. It's often worth it anyway, but not always.

• I made a spreadsheet a few days ago comparing cloak units required to get a given cloaking % and the increase in scanner range required to beat that cloaking %. XAPBob has been kind enough to host it here.
  
• The graph at the top (and another graph to the right of it) plot for every %cloak the cloak units/kT required to achieve that value and the increase in scanner range (or equivalently, scanner number in a picket line) needed to breach that %cloak. For instance, you need 420 cloak units/kT to achieve 80% cloaking, and you need 5x the scanner range to detect it, so there's a point at (420,5). The graphs only differ in that the one on the left uses a logarithmic scale.
  
• The graphs further down plot the total and marginal "efficiency" of various cloaking percentages. The total efficiency is defined as (increase in scanner power required, % of range vs. uncloaked)/(cloak units/kT required). The marginal efficiency is defined as (increase in scanner power required, % of range vs. uncloaked, compared to 1% less cloaking)/(increase in cloak units/kT required, compared to 1% less cloaking).
  
• Follow the blue lines marked "Normal" for now. I'll get to the red lines later.
  
• What can be seen from the efficiency graphs is that there are two rather different "sorts" of cloaking. There's "partial" cloaking, in which you cloak a fleet up to the 50% or 75% breakpoints for a relatively small expenditure, and there's "full" cloaking, in which you use a larger investment to go for 98% or as close to it as you can get. "Full" cloaking is far more efficient in terms of the amount of scanning required to defend equivalently against it, to the point where it becomes infeasible for much of the game for opponents to breach 98% cloaking over any significant distance (a Galleon full of Eagle Eye Scanners only achieves 11 ly range against a 98% cloak). Some awareness of these concepts already existed in the form of various vague advice and heuristics (e.g. "Super-Stealth shouldn't bother with dedicated cloaks until they can get to high 90s"), but I believe these graphs help to achieve a more complete understanding of what's going on here.
  
• Most races cannot achieve 98% cloaking prior to the acquisition of Nubians; 12 Super-Stealth Cloaks are required for a ship to reach 98%, and no other freely-available hull can mount that many (the Galleon can mount 10, but that only reaches 97% cloaking - 1.5x as visible as 98%). The exceptions are Super-Stealth races, which can achieve 98% cloaking with three Ultra-Stealth Cloaks in addition to their free 300 cloak units/kT (five Transport Cloaking units would also suffice, but no ship is capable of mounting five of them), and Hyper-Expansion races, which can mount the required number of Super-Stealth Cloaks on a Meta-Morph.
  
• Even given an appropriate hull that can mount 12 Super-Stealth Cloaks, combat ships (by which I mean fleet-battle ships, as opposed to skirmishers) are typically crippled by giving up so many slots. The generally-accepted way to get around this and cloak battlefleets is known as "overcloaking". An "overcloaker" is a high-mass ship with more cloaks than required to reach the desired cloaking level (usually 98%), which therefore has "spare" cloak units available, which can then be placed in a fleet with other designs in order to cloak the whole fleet without sacrificing slots on the other ships. It must be heavy because cloaks give a set amount of cloak units per kT of ship weight.
  
• The most effective overcloaking ship (for non-SS races) is a Nubian with 24 Super-Stealth Cloaks and 12 units of Crobmnium. Warp 10 engines are obviously helpful for fairly-pricey ships intended to accompany Nubian warship fleets, so I will only give statistics for the two Warp 10 engines worth using. Such a ship with Interspace-10 engines can overcloak 996.53 kT of fleet mass beyond its own weight to 98%; with the Trans-Galactic Mizer Scoop, 950.99 kT. (The Trans-Star 10 is not worth using on overcloakers, as the lower cost isn't worth the decrease in mass and thus power; Tritanium gives an increase in mass and thus overcloaked mass per ship, but it miniaturises based on the lowest tech and most races never research Biotechnology above 7 - the price increase outweighs the additional power.)
  
• Inner-Strength and Space Demolition races have access to very heavy components in the form of Speed Trap minelayers. These components miniaturise based on Propulsion and Biotechnology and as such are inefficient as pure overcloakers compared to their cost; however, as they actually perform a useful function (unlike armour slabs) a multi-role overcloaker/Speed Trap minelayer is still a viable design (particularly for Inner-Strength; Space Demolition may prefer to leave mine-laying to their dedicated hulls). The maximum overcloaking per ship is at 24 Super-Stealth Cloaks + 12 Speed Trap minelayers, but due to the high cost of the minelayers the maximum overcloaking per Ironium is at 30 Super-Stealth Cloaks + 6 Speed Trap minelayers. Of course, the latter lays fewer mines, even after accounting for their lower cost, so if the quantity of mines rather than just their existence is important the former is preferable.
  
• Super-Stealth races are of course capable of designing highly-effective overcloaking Nubians, but to list their designs would somewhat miss the point. Super-Stealth races have a much easier time getting to 98% cloaking, so any ship with three Electrical slots to spare can be made 98% cloaked without needing to be overcloaked, and a ship with more (such as the Stealth Bomber) becomes an overcloaker in its own right. On top of that, the amount of cloak units overcloakers must supply is reduced by everything already having 300 cloak units/kT (or more, if it uses SS' special components that cloak in addition to their normal effect) and cargo being ignored. As such, Super-Stealth rarely needs a dedicated overcloaker design at all - they prefer to make otherwise-useful ships double as overcloakers.

• First thing's first; 17 Tachyon Detectors reach the limit of 81% effectiveness of cloaking, so using more than 17 is pointless (even if there's a free slot for an 18th). However, as the whole point of using them is to defeat "full" cloaking (more on that in a sec), they do best on large ships mounting a few of both TDs and scanners, such as Galleons (and Nubians, when they appear).
  
• Now, on to the fun bit! Explaining what Tachyon Detectors actually do, particularly when considering metagame effects, was the impetus for me to start putting that spreadsheet together (and by extension, writing up this whole treatise ). The red line in the graphs is for the case where the scanner is backed by 17 Tachyon Detectors, and you'll note that it doesn't behave the same way as the blue line. Most notably, the massive spike in efficiency associated with 90%+ "full" cloaking is absent, and indeed efficiency falls off past about 75% - this is because cloaking gets far more expensive in that region to counter the massive spike in power as the hyperbola approaches its asymptote. Tachyon Detectors move the asymptote out of reach, so there's no massive spike in power and the increase in cost is passed on directly to a decrease in efficiency. Looking at it another way, reducing 98% cloaking to ~80% is a tenfold increase in detection range, but reducing 50% cloaking to 40% is only a 20% improvement.
  
• As such, while Tachyon Detectors reduce the effectiveness of all cloaking, their main function is to defeat "full" cloaking. Cloaking up to the breakpoints at 50% and 75% is still worthwhile (and as such SS isn't completely useless), but trying to get to 98% against an opponent with Tachyon Detectors is a waste of effort since it won't actually make you invisible.
  
• The only ship designs I'll provide here are those maximising range of through-98%-cloak scanning in a few different scenarios.
  
• First, the Frigate, largely for comparison. Obviously, the optimal design is that with 3 Tachyon Detectors and 2 scanners, which provides 12.3% of your scanners' rated range - an awful lot more than the 3.0% anyone else can get, but you can do better.
  
• Next, the Galleon. The Galleon has 6 General Purpose slots, 2 Elect/Mech slots, 2 Mine/Elect/Mech slots, and 2 Scanner slots (plus a couple of Shield/Armour slots, which aren't useful to detection without Mystery Trader items). The maximum scan range vs. 98%-cloaked ships is achieved with 5 scanners and 7 Tachyon Detectors, at 21.6% of your scanners' rated range (as opposed to 19.8% for 2 scanners and 10 Tachyon Detectors, or 19.4% for 8 scanners and 4 Tachyon Detectors).
  
• Finally, the Nubian. Given the large number of identical slots on the Nubian, and the likelihood in the Nubian era that you'll want extra capabilities on such a ship (cloaking, mine laying, mine sweeping...), I'll do an optimisation for each number of slots free.
  - All 12 slots: 18 scanners, 17 Tachyon Detectors (42.5% rated range)
  - 11 slots: 15 scanners, 17 Tachyon Detectors (40.6% rated range)
  - 10 slots: 15 scanners, 15 Tachyon Detectors (38.7% rated range)
  - 9 slots: 12 scanners, 15 Tachyon Detectors (36.6% rated range)
  - 8 slots: 9 scanners, 15 Tachyon Detectors (34.0% rated range)*
  - 7 slots: 9 scanners, 12 Tachyon Detectors (31.1% rated range)*
  - 6 slots: 6 scanners, 12 Tachyon Detectors (28.1% rated range)*
  - 5 slots: 6 scanners, 9 Tachyon Detectors (25.0% rated range)*
  - 4 slots: 6 scanners, 6 Tachyon Detectors (21.2% rated range)
  - 3 slots: 3 scanners, 6 Tachyon Detectors (17.9% rated range)
  - 2 slots: 3 scanners, 3 Tachyon Detectors (13.6% rated range)
  - 1 slot: 3 scanners, 0 Tachyon Detectors (2.6% rated range)
  *If running 12 Super-Stealth Cloaks to achieve 98% cloaking, and Elephant Scanners, these ships can achieve a penscanner lock on a planet through arbitrary cloaking while being safe from any power of detection on that planet (up to 36-Peerless nubian with NAS) themselves.

That is some good stuff. Would you show the how SD field scans work?

Are you sure? I am not arguing, I just seem to remember differently. Something like range instead of probability? Maybe what I remember is having this discussion before.