I went to a girls' school from age 7 to 15, then a mixed (but predominantly boys) sixth form. I don't remember huge amounts of bitchiness or bullying - but maybe that's just me.
Boys did seem like a bit of an alien species, specially since I didn't have any brothers. It did mean that when the hormones hit we didn't have boys around to distract us at school, though I have to say that when I got to the sixth form, I didn't have any trouble controlling myself around all the spotty, self-important and immature boys in my classes...
DS moved to a single-sex school for year 7, and loves it (so far). We didn't choose it specifically because it was single sex, but it was a plus rather than a negative.
One advantage is that it seems far more acceptable for boys to get involved in subjects and extra-curricular activities that would probably be girl-dominated in mixed schools - art, languages, drama, cooking competitions, iceskating lessons etc. Likewise, I've heard that in girls-only secondaries, girls are more likely to take (and do well in) subjects like hard sciences and maths which can be seen as 'boys' subjects' at mixed schools.
Also, these days most single-sex secondary schools seem to make more of an effort to encourage interaction with the opposite sex from early on - not just the discos from age 14 or 15 that we had. DS regularly has activities scheduled with one of the local girls' schools, and of course they are all in touch with each other via facebook etc, so girls aren't such an alien species for him.
I don't see so many advantages at primary level though.