You will need to install a user script manager extension to install this script.
You just have that nice button under your search (if you searched something). If you click on it, your search will get saved!
To load a search, just click on the "Saved" drop down and you'll have a list of all your saved searches, sorted alphabetically.
If you want to delete a search, you have to first click on it, that way the script knows the search was a saved one. Then, you'll see a button titled "Delete search", which you can click and your search is gone!
You might have noticed the "Dev Options" drop down on the end of the page. If you click that, you can either:
Just message me on rule34, my username is "User_314159". I'll try to get back to you there.
You can find it here, it lets you favorite artists under their posts