As far as I know, most of the wind instruments are transposing, and many of them are either in B, in E-flat or in F - these keys make them more suitable for playing in "flat" keys. My question is why there are no counterparts for playing in "sharp" keys (for example, sounding a semitone lower - in A, D and E, respectively), like clarinets, which do exist in 2 "kinds" - in B for using in flat keys and in A for the sharp ones?
The clarinet is different from almost every other wind instrument because it overblows at the twelfth, not at the octave. Therefore, the notes to cover the "break" between the lowest octave and a fifth are technically more difficult, especially on early clarinets with a limited number of finger-keys.
The orchestral use of B flat and A clarinets dates from before the instrument was redesigned following the Boehm system in about 1840. (Note, the redesign of the clarinet was not actually done by Boehm himself). Apart from the extra lowest note, most "A clarinet" parts can be (and often are) played successfully on modern B flat clarinets.
For an full ensemble of wind and brass instruments, there is no need for two "sizes" a semitone apart - most music is written in flat keys, and arrangements are often transposed up or down a semitone.
Because of the acoustic compromises needed in designing brass instruments with valves, the "best sounding" keys have a few flats more than the natural key of the instrument. For example D flat is a popular key for brass ensembles, corresponding to written keys of 3 and 2 flats for the Bb and Eb instruments.
As a relevant bit of trivia, 19th-century military bands often used piccolos in D flat because of the use of flat keys - the small physical size of the piccolo means it has fewer keys than the bigger woodwinds. This was often not explicitly marked in the score or the player's parts - and was sometimes erroneously indicated as "Piccolo in D," not "D flat".
To add to alephzero's notes on the clarinet, I'm pretty sure Adolphe Sax designed his set of instruments so a musician could easily switch from clarinet to sax and back. As it happens, there is a "C-melody sax" in rare use as well.
Now to the question of key signature: The resonances and timbres of keyed instruments depend on the bore and the bottom note (all keys/holes closed), not on the transposition. For example, the Bb clarinet's low note is E, concert D. This means one could write in a key which simplifies fingerings (which really doesn't bother anyone reasonably skilled) or in a key which produces certain sonorities.
It would take some work to provide a distribution of concert keys for which notable clarinet works exist, but I can say from experience that there's plenty both in a few flats and otherwise (including of course items in A or a for A-clarinet).