I for one have had ZERO stability issues with postgres since version 7.0, and all the issues I had with 6.5.x and before had to do with feeding it ugly sql with a 16 way where clause that made it choke, not die, just grind away for 10 minutes before returning an error.
Postgresql can handle hundreds of simultaneous transactions at a time, and seems to scale better under load than MySQL n my own testing.
They're both pretty good databases, and they both have their uses, but if you are doing financial stuff (i.e. charging) you should probably use transactions, which means you should use postgresql or maybe some other inexpensive/free database (there are others besides just Postgresql and MySQL.