Perhaps under System >> Data, have an extra button to Clear All Searches. Seems like it might fit right in there with clearing all downloads, clearing all uploads.

I delete new searches as I make them now, but over time I've accumulated hundreds or thousands of searches, and I don't see a painless way of getting rid of them (clicking the X 5 times a day, may take months...). At some point accumulating too many searches may cause the web app to lag, so would be nice to have the ability to remove them.

I also have a search that's stuck and thinks it's still in progress, I am hoping to clear that one out as well.