Wow this thread is a bit worrying for me - we've just accepted an offer at Japs at 7+ for dd, turned down Alleyn's! Pretty surprised at user's post, especially the bullying part.
Ime Alleyn's doesn't have a 'much better' reputation (especially at senior level), just a different one. Having grown up in the area and known many people from when I was at school myself, and friends with dc at both now, Alleyn's has quite a different vibe to it. We chose Japs as it seems to suit dd more, she's not particularly outgoing but pretty academic. Alleyns seems to put a lot of stock on its drama reputation, which isn't really dd at all.
I gather there were about 200 applicants for Alleyns at 7+ (18 places both sexes), about 130 for Japs (15 places).
FWIW we also loved Sydenham High, would have probably chosen it over Alleyns actually if we hadn't got Japs. Really lovely feel to it. But so much depends on your child, and it's a lot harder to tell at 4yo (as opposed to 7) what will suit them. I do think co-ed at least till 7 is better.
The main difference between SHS and Rosemead is that as Rosemead is a classic Prep it will focus on the 11+ whereas SHS (or Japs/Alleyns) won't as most will stay to 16/18.
All the big dulwich schools are ridiculously competitive,like many places in London.