When I do any database work in Java, I like to support both PostgreSQL and MySQL. The MySQL drivers since 2002 have a GPL license and are not suitable at all (for license reasons) for use in non-GPL projects when you need to distribute your application (but is fine for internal projects). Thus, for non-GPL projects, I use the older LGPL MySQL driver org.gjt.mm.mysql.Driver to stay clean, license-wise.
I do notice that just about all online tutorials that use MySQL with Java use the newer GPL licensed drivers, so I would imagine that many people end up using the GPLed driver in non-GPL projects. Being an open source enthusiast also means respecting license choices of copyright owners.
The old LGPL driver does not support some things like bit values, but since the old driver is open source, I could always tweak the source code if need be. I am a little surprised that the Java community does not fork the old LGPL driver and maintain it.