I use covers in my set to provide musical textures which my own songs do not have. I can't really do jazz chording, so I don't write in that style. However, a stripped down version of "Buddy Can You Spare a Dime?" seems to be popular, as does my (very simple) version of "Nobody Knows You When You Are Down and Out".
There are a couple of virtuosos here, who are able to play and write in blues and ragtime styles as well. Jed Marum is one. I would not feel comfortable doing that. I don't have the musical skill or cultural background.
The covers I select are more to bring something into the show that I would not or could not have written myself. I can do sing along songs, humourous songs and character songs along with ballads etc. I am never going to write anything with the grace and universality of "Somewhere Over The Rainbow" or "Singing In The Rain" though, which is why I sometimes use one of those songs to close an evening. The songs seem more effective when they catch an audience by surprise. This may sound a little arrogant, but unless I feel a song says or does something which I would not have, I do not use it.