When I was playing in jazz trios, I was expected to solo quite a bit, but it was usually a negotiation (pretty much every tune). When the time came, whoever was calling the shots would look at me and I would either nod yes or no. To be honest I won't repeatedly work in a group where this is not the way things are run. I don't have a problem with stretching myself and taking chances, but I don't enjoy falling flat on my face and I also don't like being handed only the crap solos that the bandleader doesn't want to play on, especially if he/she never looks my way during charts that I like to solo over. Maybe I am just difficult

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As far as playing the melody. That's a strategy that is recommended as a stepping stone for learning how to solo, but I don't think its a strategy you should use too frequently on a gig. Normally a tune starts of with the head, and then you don't hear the full melody again until all the solos are done. So IMHO when you hear the melody a second time it's sort of a signal that the song is coming to an end. So yeah, it can be cool, but not all the time.
If the song is so fast that all you can do it walk and someone asks to to solo...walk it brother! It will be just fine.
IMHO people have the wrong idea about just how impromptu solos actually are. Good jazz musicians spend endless hours analyzing and reviewing changes, and exploring/practicing ideas. I don't think most musicians are making up much on the spot. They are basically playing a stream of consciousness, but they are drawing on a library of licks they have worked up previously. Even if they are playing over the changes of a tune they have never played before this is true. Perhaps it's a new tune, but it's likely that the chord progression is made up of the same building blocks as some other songs they know inside and out. So IMHO the secret is listen to the greats, and study and practice a bunch of tunes....I should probably take my own advice.