337's
Whilst you can search for FAA 337's against a particular aircraft registration through the FAA in Oklahoma, you can't do a general search to see, for example, if a particular type of equipment has been installed in a particular type of aircraft. FAA 337's have to be filed with the FAA to record that a mod has been installed on that particular aircraft.
You can do a search for STC's with both the FAA and EASA, as well as AAN's with the UK CAA, EMZ approvals with the German LBA etc, but as mentioned previously, once you find who has/owns the STC, you have to pay for the package and rights to use it.
It's becoming more prevalent now for some equipment manufacturers to generate FAA AML STC's - these are Approved Model List STC's which may list hundreds of aircraft types. Garmin have the G600 EFIS on this type of approval, so if you buy the system from Garmin, you get automatic rights to use the STC when installing it. Even in EASA land, there are some AML modifications availabe - Becker have such a minor mod package for their Mode S transponders. Getting an AML STC in Europe is a different proposition, and time will tell if EASA will allow it.
Some STC holders provide a stipulation that the approval is only valid for new equipment - you can't use if for second-hand parts. This is their right as owners of the intellectual property rights. It's no different for the owner of a minor modification package. To create a database of approved minor changes would be huge. My own organisation has well over 1000 approved mods, including STC's and minor changes.