In singularities there is no more space time. Therefore there is no place for mass. Why then are black holes stable?
A quite right question, but the absence of light doesn't mean the absence of matter by any way...

Try to consider
dropplet or particle model of black hole (and Universe) by
Aether Wave theory (AWT). If we consider the vacuum composed from particles like tiny droplets, the
negative curvature of energy spreading results in strong repulsive force at short distance, thus effectively (but no definitelly) prohibiting the collapse of black hole.

Nevertheless, the high pressure inside makes the particle motion chaotic and this is why, the elastic interactions, which are able to spread some energy or information are quite limited here. You can imagine, each the particle is surrounded by nearly infinite number of neighboring particles, so that all the forces will compensate each of other at low distances. As the result, the motion of matter inside black hole becomes highly chaotic and nearly no time arrow can be observed here (the time is "reversed" or even nearly "stopped" here).
Such chaotic environment behaves exactly like the free space, which is supposed at the very beginning of Universe, so that the
AWT supposes, such environment has formed our Universe too. By such way, the Universe is formed by black holes interior recursively (see the animation on the right). Because such environment behaves like common matter, we can call it an Aether from the preEinsteinian era. As the mass of Universe increases, a big crunch of the matter appears spontaneously at the end, which can be considered as the further inflation period of new generation of Universe.