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. :)
In the manufacture of LEGO bricks the machine tolerance is as small as 10 my (0.01 mm).
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.
I have actually tried 3d printing lego on a prusa i3 printer at I think 5 microns using PLA. I found that they worked fine initially but they quickly started to lose their grip. I think the issue was the layers grind against each other which wears them down rather than the brick walls flexing to grip. You certainly won't be getting anything close to the quality of real lego but you could print some cool custom parts.
Here is a picture of the finished result of one of my brick prints: