I would like to print LEGO compatible figures. Which resolution is required for the 3D print in order to firmly fix the figures on LEGO bricks?
Depends on your tolerance. :)
http://en.wikipedia.org/wiki/Lego says: "The machines that make Lego bricks have tolerances as small as 10 micrometres."
For me, 3D printing resolution would need to be pretty close to that.
According to WikiPedia:
The moulds are permitted a tolerance of up to two micrometres, to ensure the bricks remain connected.
But, as I noted in a comment, that is in an industrial environment, where great care is taken to ensure that not just a one-off piece is right, but that the entire batch of pieces is right. They probably take a six sigma approach or something similar.
In a DIY setting, that is not necessary. A certain percentage of faulty pieces can be tolerated, as a home manufacturer probably has the time to check each piece and recycle the faulty ones.
Since the smallest offset seen so far is 0.3 LDU, you would need at least a 0.12mm resolution to print that brick. But if you're printing pieces with a resolution of 1 LDU, you'd need a printer with at least a 1 LDU resolution.
So it depends on the resolution of the pieces you want to print.