Bash and Test-Driven Development
If you are writing code at the same time with tests, try to make it high on functions that don't use anything besides their parameters and don't modify environment. That is, if your function might as well run in a subshell, then it will be easy to test. It takes some arguments and outputs something to stdout, or to a file, or maybe it does something on the system, but caller does not feel side effects.
Yes, you will end up with big chain of functions passing down some WORKING_DIR variable that might as well be global, but this is minor inconvenience comparing to the task of tracking what does each function read and modify. Enabling unit tests is just a free bonus too.
Try to minimize cases where you need output. A little subshell abuse will go long way to keeping things nicely separated (at the expense of performance).
Instead of linear structure, where functions are called, set some environment, then other ones are called, all pretty much on one level, try to go for deep call tree with minimum data going back. Returning stuff in bash is inconvenient if you adopt self-imposed abstinence from global vars...
So here is what I learned:
There are some testing frameworks written in bash and for bash, however...
It is not so much that Bash is not suitable for TDD (although some other languages come to mind that are a better fit), but the typical tasks that Bash is used for (Installation, System configuration), that are hard to write tests for, and in particularly hard to setup the test.
The poor data structure support in Bash makes it hard to separate logic from side-effect, and indeed there is typically little logic in Bash scripts. That makes it hard to break scripts into testable chunks. There are some functions that can be tested, but that is the exception, not the rule.
Function are a good thing (tm), but they can only go so far.
Nested functions can be even better, but they are also limited.
At the end of the day, with major effort some coverage can be obtained, but it will test the less interesting part of the code, and will keep the bulk of the testing as a good (or bad) old manual testing.
Meta: I decided to answer (and accept) my own question, because I was unable to choose between Sinan Ünür's (voted up) and mouviciel's (voted up) answers that where equally useful and insightful. I want to note Stefano Borini's answer, that although not impressed me initially, I learned to appreciate it over time. Also his design patterns or best practices for shell scripts answer (voted up) referred above was useful.