It depends a bit on the terminal you're in at STR.

The "designated" priority lane is at T2 and is a joke. There's a red carpet but the only priority treatment you get is that you cannot opt out of the pornographic-scanner...

T1 and T3 have "priority" stickers on the ground (the blue line) but nobody enforces anything as such it's basically non-existent.

If you talk to people at the airport they'll either point you to T2 or claim that the airport doesn't have priority-handling at all.
Personally, I think the latter group is right.

Generally though it doesn't matter. I've been flying through STR probably something like 200 times in the last 5 years (usually Friday evening and Sunday evening but a few times during the week as well) and real queues are _very_ rare.

In March 2015 somebody at the German ministry for the interior had a brainstorm though and required stricter security checks including opening and switching on of all electronic devices etc.
Queues went up to 45min and longer just at security.
This stopped after 4 days though, not sure why exactly.

But generally, STR is a quite nice airport with very short queues...