Singaround layouts and formats vary:

a) Some put the chairs in a circle (I don't like that, myself, too "twee")
b) Some have the floorsingers get up onto a "stage" area
c) Some just do it from where they are sitting (my preferred style)
e) Some take it in turns to do 1 number on a roundrobin basis left to right or the reverse
f) Some are MC-ed (the MC has a list of names and asks the next floorsinger to be getting ready as the previous one is about to start). If the MC is good, this can be a very good way of mixing styles and quality.
g) Some do 1 song at a time; others do 2 at a time.

Usually at a singaround you just turn up and if you are new to the club let organiser that you are there and would like to participate.

Some are weekly, some are fortnightly, some are monthly. Then there are impromptu ones (we just ran one because Dan McKinnon was in the area and had a night free).

Usually the singarounds are not themed - but some clubs like to have a themed night once in a while. Personally I don't like that, I find it stifles spontaneity - better to take my cue for what to sing from the person that sang before me.