Does electron in an $s$-orbital have angular momentum? I've read on science forum that electron in orbital $s$ has no angular momentum and would fall into nucleus, so hydrogen atom would not be possible. Electron has mass and speed after all right? So how is it possible?
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I have read on science forum that electron in orbital s has no angular momentum and would fall into nucleus, so hydrogen atom would not be possible.

In this sentence you are encapsulating the reason quantum mechanics was "invented". The planetary-like theory of the Bohr model imposed the "not falling", quantized stable orbits with a minimum binding energy, to explain this paradox.  With the hydrogen solution of the Schrodinger equation, quantum mechanics came into being as a theory for the microcosm; the stability of the energy levels is basic in its postulate. Quantum mechanics has been verified over and over again as the theory that describes the microcosm of atoms and elementary particles.

Electron has mass and speed after all right? So how is it?

An electron has mass and speed wherever we can measure its momentum. Bound in the atom we cannot measure it without kicking it out of its energy level; we rely on the probability functions given by the quantum mechanical solutions to describe the probable position it will have. As the position is only known by a probability function, an angular momentum vector cannot be assigned, and speed is only known after the measurement. They have managed to "see" the orbitals of the hydrogen atom at its points of interaction.
The s-orbital electrons do have a probability of going through the nucleus, as has been measured by electron capture nuclear reactions.  It does not happen often because in the quantum mechanical framework a number of other considerations enter in the solution, such as the existence of proper energy levels and quantum number conservation. In any case, there is a factor of 10^5 between the radius of atomic orbitals and the nucleus, thus further reducing the probability of interaction.
A: Electron has mass and momentum, but we don't know anything about how the electron "moves" between the various points within that orbital. Stated another way, we can observe that the probability of finding the particle around the nucleus has a particular form (the s-orbital), and the "motion" if it could be defined for a wavelike dispersed electron in an s orbital, is such that the rotational angular momentum is zero. It doesnt mean it is static and will fall into the nucleus. Remember that this is not classical mechanics and the "solar system" analogy usually does not work.
