First of all, when it comes to distinguishing 3/4 from 3/8, it is important to know that there are no sound differences between the two. Also, 3/8 is almost never used because there is 3/4. The only time you would see 3/8 is maybe in a 6/8 song, where they throw in a 3/8 measure.
Now, when it comes to distinguishing the time signature, it is important to understand the strong/weak beat patterns:
2/2 or 2/4: For each measure, the first beat is strong, and the second beat is weak. So it follows this pattern: S W.
3/4: S W W
4/4: S W M W (the M is a medium-strong beat, meaning stronger than the weak beats, but weaker than the strong beats).
5/4: Either S W W M W (most likely scenario) or S W M W W
6/8: S W W M W W
7/4: Either S W M W S W W (most likely scenario) or S W W S W M W (basically it either sounds like 4/4 with a 3/4 measure following each measure, or 3/4 with a 4/4 measure following each measure)
8/4 does not exist.
9/8: S W W M W W M W W (it may sound like 3/4 where each beat has a triplet)
10/4: S W W M W W S W M W or S W M W S W W S W W (basically a combination of two 3/4 measures and one 4/4 measure, because 3 + 3 + 4 = 10)
I have never seen a song in 11/4, nor have I even seen it written in advanced music theory. If a song is ever in 11/4, you would just be able to tell because of how weird and messed up the song sounds.
12/8: S W W M W W MS W W S W W (MS is stronger than the medium beat, and weaker than the strong beat; also, 12/8 is rarely used; composers normally use 6/8 with the exception of Chopin in his Nocturne in Eb Major, Op. 9 No. 2)
When it comes to distinguishing between 2/4 and 4/4, I look at it like this:
If the song has a drum set, it makes it more obvious. If it follows the pattern of kick-snare-kick-snare-kick-snare..., it's in 2/4 (or 2/2). If it adds some sort of articulation to the second kick (like if it counts like: 1 2 3 + 4), it's in 4/4. Usually, the kick would be on the upbeat of 3, or the high-hats may have some sort of articulation.
If it has no drum set, then you try and feel out the groove of the song. Usually, the third beat of 4/4 isn't much stronger than the weak beats. So, it would feel like this: One Two Three Four One Two Three Four.... However, 2/4 feels like this: One Two One Two One Two....
The difference between 3/4 and 6/8 are very similar to the difference between 2/4 and 4/4. A drumbeat would go One Two Three One Two three in 3/4, and in 6/8 it would go One Two Three Four Five Six One Two Three Four Five Six..., usually adding a snare on beat four. If there's no drum set, then you would still feel the groove of the beat (or the swing of the beat).
6/8 and 12/8 have very slight differences. In 12/8, the seventh beat is only slightly stronger than the rest of the beats, but it's not as strong as a strong beat. Usually, a melody will match up to either six or twelve beats. If the melody repeats or can repeat below twelve beats (so basically six beats), it's 6/8. If the melody repeats or can repeat only on multiples of twelve (mostly just 12 beats), than it's 12/8. Keep in mind that 12/8 is far less common than 6/8.
The difference between 3/4 and 9/8 is simple: if each beat in 3/4 is straight (so it uses quarter notes, eighth notes, sixteenth notes, or the like), then it is simply 3/4. If each beat is a triplet (or has a triplet sound, meaning it sounds like there is a triplet on each of the three beats), then it's 9/8.
There is no sound difference between 6/4, 6/8, and 6/16. However, 6/8 is used far more commonly than 6/4 or 6/16. I highlighted it below:
2/2 2/4 2/8 2/16
3/2 3/4 3/8 3/16
4/2 4/4 4/8 4/16
5/4 5/8 5/16
6/4 6/8 6/16
7/4 7/8 7/16
There is no such thing as 8/4, 8/8, or any measure with eight beats.
9/4 9/8 9/16
10/4 10/8 10/16
Like I said before, I don't know if 11/4 exists as a standalone song. If anything, an 11/4 or 11/8 or 11/16 measure is added in the middle of a song for an articulation effect. If anyone knows of this definitely, then please tell me because it's very interesting!
12/4 12/8 12/16
There are never any occurrences of 13/4, 14/4, 15/4... that make it common enough to take note on, besides the couple of songs out of the millions and billions of songs in the world. Basically, never ever worry about these time signatures.
I hope this helped, James KI, and all other people who struggle with this issue!
Edit: The Mario Kart 64 Result 1 song (the song when you win) is in 11/8 time. So, I guess they do use it.