I am not quite sure what you mean. There is tons of experimental evidence for quantum mechanics.
Specifically for electron spin, you seem to agree that it has spin 1/2 (because you say that it can be spin up or spin down). But that already concedes that there are superpositions of up and down. Spin left, spin right, spin straight forward, spin backward are all equal superpositions of spin up and spin down; and all these states are trivially constructed.
Bound electrons (in an orbital) can only have 1 of 2 quantum "spin" states - which is in relation to the nucleus and/or the magnetic field. Up or down. With the spin-up electron having a slightly higher or lower energy level then the spin-down electron (depending on other factors).
I think the question is how you unambiguously measure spin-right or spin-left as existing. Most experiments report only a stream on spin-up and spin-down measurements, inferring the original superposition.
Specifically for electron spin, you seem to agree that it has spin 1/2 (because you say that it can be spin up or spin down). But that already concedes that there are superpositions of up and down. Spin left, spin right, spin straight forward, spin backward are all equal superpositions of spin up and spin down; and all these states are trivially constructed.