Hi,
Technically speaking there could very little difference between the three database engines you have mentioned. Take a look at this
link.
My recommendation are as follows...
1. MySQL widely used across and among the three it is most popularly used and is ranked as No 2. We have used MySQL/DCM4Chee for over million studies/images and have no problem. Works on various platforms, Windows, Mac, Linux.
2. PosGreSQL, when you need low foot print compared to MySQL this could be an option. Takes Lesser Memory and equally robust.
3. FireBird, Inherited from Interbase from Borland products - very robust engine. From Popularity point of view could be the least among your list. I have not used it but have a good review on the internet.
Hope this helps...
with regards
Rady