Maybe I should put the question this way: is or is not PL topology an integral part of the education of every geometric topologist today?
According to a recent poll by the Central Planning Commitee for Universal Education Standards, some geometric topologists don't have a clue about regular neighborhoods, while others haven't heard of multijet transversality; but they all tend to be equally excited when it comes to Hilbert cube manifolds.
some recommended references (textbooks) for a beginner
Rourke-Sanderson, Zeeman, Stallings, Hudson,
L. C. Glaser, Geometrical combinatorial topology (2 volumes)
Is there any more modern textbook on the subject?
Not really, but some more recent books related to, or with chapters on, PL topology include:
Kozlov, Combinatorial algebraic topology
Buchstaber-Panov, Torus actions and their applications in topology
and combinatorics
Matveev, Algorithmic topology and classification of 3-manifolds
2D homotopy and combinatorial group theory
Buoncristiano, Rourke, and Sanderson, A geometric approach to homology theory
(includes the PL transversality theorem)
The Hauptvermutung book
Buoncristiano, Fragments of geometric topology from the sixties
Daverman-Venema, Embeddings in manifolds
Also I would like to know some important open problems in the area, in what problems are mathematicians working in this field nowadays
I'll mention two problems.
1) Alexander's 80-year old problem of whether any two triangulations of a polyhedron have a common iterated-stellar subdivision. They are known to be related by a sequence of stellar subdivisions and inverse operations (Alexander), and to have a common subdivision (Whitehead). However the notion of an arbitrary subdivision is an affine, and not a purely combinatorial notion. It would be great if one could show at least that for some family of subdivisions definable in purely combinatorial terms (e.g. replacing a simplex by a simplicially collapsible or consructible ball), common subdivisions exist.
See also remarks on the Alexander problem by Lickorish and by Mnev,
including the story of how this problem was thought to have been solved via algebraic geometry in the 90s.
2) MacPherson's program to develop a purely combinatorial approach to smooth manifold topology, as attempted by Biss and refuted by Mnev.
