B-flat or C Trumpet for Solo Part?

Which one is more common (especially for soloists)and what are the differences between them (other than the slightly different range)? What transposition is best for a solo part?

Bb is the more common one.

What do you mean what transposition is best for a solo part?

It just transposes like a regular instrument (the Bb one that is).

So you just go about writing for it that way. It's no more difficult than writing for a clarinet etc.

I don't think there are any differences, in terms of sound or technique. No big ones anyway. I think fingerings might be different when playing in different keys, but a good player should be equally able in all keys.