How to detect and debug multi-threading problems?

Threading/concurrency problems are notoriously difficult to replicate - which is one of the reasons why you should design to avoid or at least minimize the probabilities. This is the reason immutable objects are so valuable. Try to isolate mutable objects to a single thread, and then carefully control the exchange of mutable objects between threads. Attempt to program with a design of object hand-over, rather than "shared" objects. For the latter, use fully synchronized control objects (which are easier to reason about), and avoid having a synchronized object utilize other objects which must also be synchronized - that is, try to keep them self contained. Your best defense is a good design.

Deadlocks are the easiest to debug, if you can get a stack trace when deadlocked. Given the trace, most of which do deadlock detection, it's easy to pinpoint the reason and then reason about the code as to why and how to fix it. With deadlocks, it always going to be a problem acquiring the same locks in different orders.

Live locks are harder - being able to observe the system while in the error state is your best bet there.

Race conditions tend to be extremely difficult to replicate, and are even harder to identify from manual code review. With these, the path I usually take, besides extensive testing to replicate, is to reason about the possibilities, and try to log information to prove or disprove theories. If you have direct evidence of state corruption you may be able to reason about the possible causes based on the corruption.

The more complex the system, the harder it is to find concurrency errors, and to reason about it's behavior. Make use of tools like JVisualVM and remote connect profilers - they can be a life saver if you can connect to a system in an error state and inspect the threads and objects.

Also, beware the differences in possible behavior which are dependent on the number of CPU cores, pipelines, bus bandwidth, etc. Changes in hardware can affect your ability to replicate the problem. Some problems will only show on single-core CPU's others only on multi-cores.

One last thing, try to use concurrency objects distributed with the system libraries - e.g in Java java.util.concurrent is your friend. Writing your own concurrency control objects is hard and fraught with danger; leave it to the experts, if you have a choice.

I thought that the answer you got to your other question was pretty good. But I'll emphasis these points.

Only modify shared state in a critical section (Mutual Exclusion)

Acquire locks in a set order and release them in the opposite order.

Use pre-built abstractions whenever possible (Like the stuff in java.util.concurrent)

Also, some analysis tools can detect some potential issues. For example, FindBugs can find some threading issues in Java programs. Such tools can't find all problems (they aren't silver bullets) but they can help.

As vanslly points out in a comment to this answer, studying well placed logging output can also very helpful, but beware of Heisenbugs.