We have installed the Magento 8788 security patch on several of our customer's sites and here's what we think needs testing, due to the changes the patch is making:
* Check that the image uploader works OK. Note that the image upload thing breaks if you have ssl enabled on the backend but it's just http!
* Check that you can delete the customer address in the frontend account section
* Check that you can delete an item from the wishlist
* Do a test checkout with all the payment methods (Paypal express has been changed)
* Any outgoing SSL connections will now verify peer cert by default, but this testing probably covered by testing the payment methods
* Make sure you check the max dimensions of images in SC-Catalog->Product Image->Maximum resolution for upload image. Change to big like 10000.
* Downloadable products have been affected, test them if they exist
* Test image uplaoder works on CMS admin
* Test edit cart item
* Check dashboard still works especially on 1702.
* Check products add to cart works on product page
* Check you can update the qty of items in the cart
* Check you can checkout! Form key added to review/info.phtml
* Check customer can login on both the checkout and customer account
* Check you can add a review from the product page
If you can suggest any more areas that need testing please get in touch and let us know, hope this helps.