Flight history of the Space Shuttle Main Engines

List of sources, roughly in order of most to least trusted

1:

NASA/TM–2011–216142

This is a Space Shuttle Missions Summary published by NASA in 2011 on conclusion of the STS program.

This is by far the most complete source publically available to my knowledge.

2:

The exact origin of this site is beyond me, but it is hosted on the NASA.gov site so I consider it trusted.

This was the first source I found and is complete up to STS-107. Another site is linked for that info, but the link is not available any more.

3:

File:SSME Flight History.png

This is an image from the Wikipedia article on the RS-25, based on physical printouts handed out at the KSC Press site by Pratt & Whitney

This image is also the only one to show the Powerhead serial numbers as separate to the regular engine number. The discussion over just how much of an 'engine' is considered the engine is a discussion for another day

4:

CollectSPACE

In 2006 user Paul78zephyr asked the same question I have set out to answer. Sources 1 & 3 were found on the CollectSPACE forums

However the exact serial numbers are rather disorganised

Data collected and presented by @Siwelthelongboi