It just seems like one day they had a search function, next they didn’t. What happened?
Works for me.
I never found their own search worked very well.
On the other hand, searching google for ‘tv tropes [terms]’ always seems to do the trick. 
It works for me. I got a “Can’t connect to database” type error a few days ago though; perhaps it was down when you tried.
There used to be two search functions, a google based one that searched the whole site, and another built in one that just searched for tropes, but that one seems to be gone now.
OK; I went there today and the search function was different. You had to go to “Tools”, call up Search, and then type in one of those strings of letters from an image. There was a message saying that I wouldn’t have to do that again once I became “known”; when I went again search was normal.
My guess is that the site was having problems from oversearching and the search function was getting crashed. So they put in a speed bump of sorts.
Uh, while it’s possible I have some sort of weird cookie that’s causing the search to show up, it’s -right- -there- on the front page. I just tried it in IE, which I never use, and the search field was right where it always was - below the “page history” link. The only thing that’s changed is they ditched their internal search engine for a google powered one.
And that’s what I got the second time I went there today.