"Most popular" is subjective. There are many genres of music and over 800 years of music written in a form we'd recognize as music, though it wasn't until the 14th century that French Catholic copyists would develop the timing system most similar to what we have today, which then spread across the Roman Catholic world and across to the Americas. Even then, what we now use in the "West" is as odd-sounding to Middle Eastern, African, Indian and Chinese ears as their traditional styles are to us.
But I digress.
The single most common time signature for the last 100 years has been 4/4 time, so called "common time". 90% or more of "popular" music in the last century has been in this time signature, as well as the majority of orchestral, choral and military band music.
The next most common is probably 3/4, popularized by the German waltz and its subsequent influence on Baroque composers.
After that, it gets a little hazier. The next most common, IMO, would be "cut time", 2/2. Visually, music in 2/2 time is identical to 4/4, but the feel is different; at the same indicated tempo, the measures go by twice as fast because the half note now gets the beat. Similar in timing but with more familiar note divisions to modern eyes, 2/4 time has much the same feel, but halves the written note values. Both of these are common march time signatures.
After that I would think that the "triplet" time signatures would be next. These are a descendant of "mensural" time notations, which instead of telling the performer how many of what type of note comprise a measure (because there were no measures), they told the performer how many of the next lower note value a particular note could divide into. As such, a mensural time signature of 3/1 indicated that a breve was divided into three semibreves instead of the usual two that we typically think of. The feel of this timing was preserved by using 8th-note-based time signatures with a multiple of 3 in the numerator, like 3/8 (one triplet "beat" per measure), 6/8 (two), 9/8 (three) and 12/8 (four). If I had to rank these in order, 6/8 is probably the most common, followed by 3/8, 12/8, then 9/8.
Finally, relatively rare but unmistakable, are "unfinished" time signatures, where the number of beats isn't even or a multiple of 3. The most common of these is 5/4, the most famous examples being Holst's "Mars" movement from "The Planets", and Dave Brubek's "Take 5". The original theme to "Mission Impossible" is in a similar 10/4 time. 7/4 is not unheard of but very rare, often split into an "alternating" time signature like 4/4+3/4 or even 5/4+2/4. The feel of such signatures leads into the next measure faster than you'd expect, lending an unresolved, ever-hurrying quality to most songs written in them.
Most other time signatures are relative curios. Theoretically, you could make any time signature you wanted to fit the feel of the song you wanted to write, as long as the "denominator" of the signature was a power of two (1,2,4,8,16,32,64, etc). However, with a few exceptions, it's generally encouraged to notate the song in a "lowest-terms" notation; you can write in 8/8, but it's almost as easy to write the same thing in 4/4, and much easier to read. Along the same lines, I think the conductor of a group to which you brought a piece written in 123/128 time would beat you over the head with his stand for wasting his time.
Finally, you don't even have to write in a time signature. As I said, early Renaissance composers didn't even know what they were, using a completely different system. Some modern composers have eschewed it as well for at least some of their pieces, such as Sir John Tavener's "The Lamb". It's often conducted in such a way as to have a more regular feel, but in fact the 20 measures of this piece as originally set down are a mix of 11/8 and 8/8 with a couple 11/4 measures thrown in, none of it explicitly notated as such.
So, the list:
6/4 (seen as a half-time version of 6/8 or a double-length 3/4)
It's not 15, but if you have these, and the ability to alternate or switch between them, users of your app will likely never ask you for more.