Is the feedback vertex set problem on planar bounded degree graphs hard? Is it known whether the feedback vertex set problem on undirected planar graphs of bounded degree is $\mathsf{NP}$-hard?
According to Garey and Johnson's book Vertex Cover is NP-complete on planar graphs of maximum degree four. Using a simple reduction from Vertex Cover to Feedback Vertex Set should give maximum degree eight and preserve planarity.

VC to FVS: Replace each edge by a triangle (or a double edge).

One note:
Garey and Johnson also state that directed FVS is NP-complete on planar digraphs with no in- or out-degree exceeding two. They do not specifically mention undirected FVS under such restrictions.