I know that different applications have different database requirements but I still like to have standard tools that I use for most jobs.
For many years the words "database" and "PostgreSQL" were synonomous for me :-)
Still, my usual deployment pattern is to help a customer rent a cheap dedicated Linux server and set it up for them so that it supports the web application that I wrote for them.
PostgreSQL is not often installed out of the box in leased Linux servers, so using it is a little extra overhead. MySQL is almost always installed, but I really don't get the same joy from MySQL that I get from PostgreSQL.
I have used HSQLDB (an embedded Java database), but little problems like sometimes locks not getting reset have kept me from using it in production environments.
I am experimenting with using IBM Cloudscape (or the open source Derby version) as an embedded database in Tomcat (using Tomcat server events to support smooth Cloudscape initialization and shutdown). I am using this for my own web portal (a business for my wife and I) before trying it on any customer jobs.
Ideally, I would like to have one standard bundle of Tomcat + an embedded database server that is preconfigured for fast deployments to new servers.