According to this site, Maine, Massachusetts and Vermont allow current inmates to vote. This article, perhaps more recent, adds New Hampshire to the list.

As for convicted felons who've served their sentences, there is a process in most states where they can regain their voting rights. According to this article from 2000, at least 10 states prohibit convicted felons from regaining their rights. This is based on a clause in the 14th Amendment.

The issue of convicted inmates voting is distinct from inmates who are awaiting trial and have not been convicted. Some jail inmates couldn't make bail but have not been convicted of a crime, so they have the right to vote.