It's not entirely clear whether the original question relates to the UK or the US (often the cause of much confusion on Mudcat).

I can't speak for the US, but I'd say that in the UK the organisers will usually make it clear what their general policy is (you can always ask them if it's not apparent to you). It's not necessarily a case of narrow-minded "purists" getting in the way of good music - some clubs activly encourage certain types of music, so everyone knows what they're getting and can go elsewhere if it's not what they're looking for.

If a club actually calls itself "open mic", that would suggest to me that most of the music will be instrumental or accompanied, with less audience participation than at a singaround style of club. You'd probably be fine there, whereas a solo unaccompanied singer might not be so comfortable. If on the other hand they say it's a song club, they'll be more keen on songs that people can join in with, and much of it may be unaccompanied - you might not fit in so well there. Visit the club and you'll soon be able to work it out.

Sessions are sometimes specific about the genre or origin of the music (eg only Irish, etc) but clubs are generally happy enough to welcome varied contributions - their main problem is not the choice of material but how to discourage the really bad performers. People who play as a duo have generally practised together and are thus good enough to entertain others (which is plainly not the case for all solo performers!), so I'd think the right sort of club would welcome you.

Marje