We do have a "Give us feedback" link at the bottom of each search results page which does some neat AJAX-y things. But even then, we get a lot of "Hi, I'm trying to find my great-great-great-grandfather. He last name was Smith. Thanks!" sort of submissions. Also submissions like "My computer keeps humming. I opened it up and cleaned out the dust, but it still hums. Do I need to clean out the dust again?" And some submissions that say "I would like to rank #1 for all my keywords. How do I do that?"
I've wondered whether a Chrome extension or something similar would give higher quality data for spam or bug reports. It might be worth thinking about offering something like that.