Both. Some don’t.
The angular momentum for the electron in the hydrogen atom is zero. This means that the electron is not moving around. (For experts: I assume that the atom is in its ground state.
What is the orbit of it? The wave function doesn’t change with angle. It can be thought of as orbiting in all circular or elliptical orbits, including those that pass through the nucleus, at the same moment. This is not how I see it. I don’t believe the simple wave function contains all the information. However, it may be useful to try to understand it using classical physics.
Some atoms have net angular momentum. This can be used to determine a sense for rotation. This sense can be either counter-clockwise or clockwise, or any combination thereof. To make an atom behave one way or another, you can apply forces to it (e.g. It can be made stable by applying a strong magnetic field. The relative spins of atoms bound to each other can be fixed. However, an atom can have its own angular momentum in a completely random direction.