The amount of units you can move on a transport can be quirky when you are moving anything other than T1 units. I wanted to put together a complete list of all the rules. So I sandboxed it. I would appreciate any additions or corrections.
First - I am only looking at what can happen in a normal game. I know there are some mods or special maps, like survival games, where there are transports carrying T4 units. I'm not interested so much in what the engine is capable of, but what players are capable of in a FAF game (1v1 ladder or global ranked). I'm not interested in looking at mods like Equilibrium to find out if the rules are all the same there as well.
Second - the rules about what can be transported are very straightforward. It appears that every land unit, other than T4s, is transportable. Structures, naval units, and air units are not transportable. (I'm not looking at which air units fit into a carrier.)
Third - the Stinger (the UEF T2 gunship) is a special case. It can carry exactly 1 T1 unit or exactly 1 T2 unit.
Fourth - there are basically 3 classes of transportable units: T1, T2, and T3 land. Transports can carry more T1 than T2, and more T2 than T3. SACUs are T3 units. ACUs are treated as T3 units. There is just one exception: Fire Beetles are treated as T1 units. Other than fire beetles, every T1 land unit takes up the same amount of space on a transport as every other T1 unit. Every T2 land unit takes up the same space as every other T2 unit. And every T3 unit takes up the same amount of space as every other T3 unit.
Fifth - there is a special rule against T1 transports being able to carry ACUs. All T2 and T3 transports can move ACUs, and they take up exactly the same space as any other T3 unit. (The rule banning ACUs from T1 transports is purely about game balance - the T2 and T3 transports all treat ACUs exactly the same way as they treat all other T3 units.)
Sixth - except for Stingers, EVERY time you can fit a T2 unit, in its place you can fit 2 T1 units instead. ALWAYS.
Seventh - EVERY time you can fit a T3 unit, in its place you can fit 2 T2 units. ALWAYS. (And because you can fit 2 T2 units, you could instead fit 4 T1s, or a T2 and 2 T1s.)
Eighth - but the reverse is not always true. Sometimes you can fit 4 more T1 units, but not a T3. Sometimes you can fit 2 more T2 units, but not a T3.
So with that out of the way:
With the exception of Aeon transports, it appears there are 4 types of "docking clamps" that transports can have, and every transport has some combination of these clamps:
A - Stinger-type clamp - this can hold any 1 T1 or T2 unit.
B - Small-type clamp - this can only hold 1 T1 unit.
C - Medium-type clamp - this can hold a T2 unit, or 2 T1 units.
D - Large-type clamp - this can hold a T3 unit (or 4 T1s or 2 T2s or 1 T2 and 2 T1s) (on a T1 transport, a large claim cannot pick up an ACU)
Here is a list of non-Aeon transports. First, the number of mech marines they can carry; second, their faction; third, their name; fourth, their clampage.
Gunships
1 - UEF - Stinger - has 1 Stinger-type clamp
T1 Transports
6 - UEF - Courier - 1 large + 2 small
6 - Cybran - Skyhook - 1 large + 2 small
8 - Seraphim - Vish - 1 large + 2 medium
T2 Transports
14 - UEF - Star Lifter - 3 large + 2 small
10 - Cybran - Dragon Fly - 2 large + 2 small
16 - Seraphim - Vishala - 4 large
T3 Transports
28 - UEF - Continental - 6 large + 4 small
You will notice that the UEF T3 transport can carry exactly the same as 2 UEF T2 transports.
Aeon:
All of these rules and exceptions that I listed above are just too simple and straightforward. Obviously, a game as rich and deep as Supreme Commander needs to have more complicated rules. That's why we have Aeon transports.
The Chariot (Aeon T1 transport) can carry all of the same things as a UEF or Cybran T1 transport. (a T3 + 2 T1s, 2 T2s + 2 T1s, 1 T2 + 4 T1s, or 6 T1s).
BUT it can also carry 3 T2 units for some reason. HOWEVER, it cannot carry a T3 unit + a T2 unit.
So the Chariot behaves like it has 3 medium clamps OR it can behave like it has 1 large + 2 small clamps.
The Aluminar (Aeon T2 transport) is similarly convoluted.
It can behave like it has 2 large + 4 small clamps. (It can move 2 T3 + 4 T1s. When it has 2 T3s loaded, it cannot pick up any T2s. But when it has just 1 T3 loaded, it can pick up 2 more T2s and 4 T1s.)
Or it can behave like it has 6 medium clamps. (It can move 6 T2 units, or in place of any of those T2 units, you could put in 2 T1s.)
2 Chariots can move all of the things that 1 Aluminar can move, EXCEPT of course they can't move an ACU.
1 Aluminar can move all of the things that 2 Chariots can move, EXCEPT that 2 Chariots can carry 1 T3 + 3 T2 units. And an Aluminar can't do that.
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Based on these clamp rules, the easiest transports to use should be the Vish, Vishala, Star Lifter, and Continental. Out of all the transports, they have the least amount of weirdness if you're trying to carry combinations of units.
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An additional note about clamps and load order:
The order in which you load units matters. For example, if you have a UEF T1 transport (courier), it can pick up a titan and 2 mech marines. But if it picks up one mech marine, it can't pick up a titan. Because I guess the first mech marine that gets picked up goes straight into the large clamp. And that means, the order you unload units also matters. (If you load a titan and 2 mech marines, you can drop off the titan, and then pick it back up - so it's not like there is a rule that "whenever a courier has 1 mech marine onboard it can't pick up a titan" - it is just a matter of which clamps are blocked)
This can even happen when you are loading an empty transport. I put units physically in a line in this order: lab - deceiver - lab - SACU - lab - deceiver - lab. Then I selected all of them and gave a load order to an empty aluminar. The Aluminar loaded the deceivers (T2) first, then the labs (T1), and didn't have enough room for the SACU (T3). So I hate to tell you this, but in some cases you might have to micro-manage the order in which your transports load units. You would generally want to load T3 first, then T2, then T1.
You can't trust the game engine to load units in a smart order. It would be smart if the first T1 units you picked up with a courier went straight into the two "small" clamps. It would be smart if the Aluminar picked up the T3 unit before it loaded T2 and T1. But the game engine (or at least, the current implementation) is not so smart. I'm not sure it's worth trying to "fix" this by changing the order in which units are loaded into a transport, because that could cause other problems (if transports don't try to load the closest units first, they might take an inefficient/overly long route to pick up all of the units). However it WOULD be nice to have a fix where transports load units by putting them onto the smallest available clamp first. I have no idea how hard that would be to implement. Right now it seems like the transports just have a list of clamps and always try to load units in order, and this list starts with the largest clamps and ends with the small ones. Just reversing that list (trying to load the smallest clamps first) might help. If you're going to put T1 units on a continental, it makes sense that they would go into the T1-only slots first, before you start putting them in slots that could hold T3 units.
Here is another example of how the order can matter.
Make a chariot. Pick up 6 t1 units, in this order: tank - lab - tank - lab - tank - lab. Now drop off all of the labs, but not the tanks. You will have a chariot carrying 3 tanks. So it should have enough room to pick up a T2 unit, right? Because we know that a Chariot can pick up 1 T2 + 4 T1, if it wants, and it only has 3 T1s onboard. But if you try it, it won't work. Because each of the 3 "medium" clamps has 1 T1 unit on it. You would have to unload the tanks first. Then you would be able to load a T2 tank + 3 T1 tanks. This I don't think we need to fix. Teaching transports to shuffle units between hard points, or teaching them to automatically drop off and re-pick-up units in order to make more room seems like it's (a) too complicated to code and (b) is likely to introduce bugs and other undesired behavior (c) compared to the value it would add to the game. So I think we have to just live with this particular quirk of the game.