Which quantum number distinguishes the different shapes of the orbitals?

The four quantum numbers are:

principle quantum number: this number describes the energy of orbitals. It describes the most probable distance between the electron and the nucleus.

angular quantum number: this number describes the shape of orbitals, and thus, describes the angular distribution.

magnetic quantum number: this number describes the number of orbitals and how they are oriented within the subshell

spin quantum number: this number determines the direction of the spin of the electron.

Based on the above, the quantum number that distinguishes the different shapes of the orbitals is the angular quantum number