Many fundamental issues remain for black hole thermodynamics after almost
half a century of its conception. For example, what are the underlying degrees
of freedom of a black hole horizon that give rise to said thermodynamical
properties? Furthermore, classical black holes also harbor a spacetime
singularity. Although it is often believed that quantum gravity would "cure"
the singularity, as emphasized by Penrose, this viewpoint requires a deeper
examination. In this review, I will examine the possibility that singularities
remain in quantum gravity, the roles they may play, and the possible links
between singularity and black hole thermodynamics. I will also discuss how --
inspired by Penrose's Weyl curvature hypothesis -- gravitational entropy for a
black hole can be defined using curvature invariants, and the surprising
implication that the entropy of black holes in different theories of gravity
are different manifestations of spacetime curvature, i.e., their underlying
microstructures could be different. Finally, I review the "Hookean law"
recently established for singly rotating Myers-Perry black holes (including
Kerr black holes) that connect black hole fragmentation -- a consequence of the
second law of black hole thermodynamics -- with the maximum "Hookean force", as
well as with the thermodynamic geometry of Ruppeiner. This also suggests a new
way to study black hole microstructures, and hints at the possibility that some
black holes are beyond the Hookean regime (and thus have different
microstructures). While examining the remarkable connections between black hole
thermodynamics, spacetime singularities and cosmic censorship, as well as
gravitational entropy, I shall point out some subtleties, provide some new
thoughts, and raise some hard but fundamental questions, including whether
black hole thermodynamics is really just "ordinary thermodynamics" or something
quite different.