[Sticky] PlayStation Links & Resources (PlayStation 3, PlayStation 4, PSP, Vita)
Check out the general research forum for more useful links and resources that are cross-platform. The following will help you dig up information on PlayStation titles only:
PlayStation.Blog Search Page (North America)
History of new game releases, sometimes with interviews and posts from the developers. History of game sales, useful for narrowing down the timeframe of a delisting (usually just after a sale ends). Uses flickr to host a lot of their images so there's a huge history of reasonably high quality uploads you can dig through just on their flickr account.
PlayStation.Blog.Europe - (Europe/SIEE region)
Mostly the same as their North American counterpart. I use them mostly to find when a delisted game was last on sale to help narrow down a timeframe.
It's hit or miss here. The Support Library has some useful pages with dates and timelines for online service disconnects. There are some specific game-related pages that have dates or details on when online play or other online features were turned off. The Forums are an even bigger crapshoot. I usually rely on GameFAQs, PlayStationTrophies, or reddit but sometimes the official PlayStation forums will have a post right around the time a game was delisted.
PlayStation Support Library (North America)
PlayStation Support Forums (North America)
PlayStation Network Store
Here are links to the majority of PlayStation Network pages organized roughly by region. Browsers (especially Chrome) will sometimes default you to your local region. You can tell by looking at the URL to see if it changed from "/en-GB/" to "/en-US/" for example. Try using Incognito or Privacy modes in your browser or a tracking-free browser.
North & South Americas
There are a few sites I also use to check on PlayStation delistings. PlayStation.com has a page for thousands of titles that were ever released going all the way back to PlayStation 1. The vast majority of these pages are not removed when a game is delisted so it's useful for finding the link to its PSN page or grabbing some details about the game. I'm not sure if the quantity is the same outside of North America though but here are the European and Japanese equivalents.
Psprices.com used to have pages for more delisted games. It was good for grabbing some cover art for the site if nothing else. They've refreshed their database and a lot of the old data is gone.