Have there been any experiments performed (or proposed) to prove that the shapes of the s,p,d,f orbitals correspond to our spatial reality as opposed to just being a figment of the mathematics that give us something to visualize?
A few years ago the XUV physics group at the AMOLF Institute in Amsterdam were (to my knowledge the first to be) able to directly image the orbitals of excited hydrogen atoms using photoionization microscopy. For more details see the paper,
Hydrogen Atoms under Magnification: Direct Observation of the Nodal Structure of Stark States. A.S. Stolodna et al. Phys. Rev. Lett. 110 213001 (2013).
This was actually featured as one of Physics World Top 10 Breakthroughs of the year 2013. There is a nice open access Viewpoint on this if you want to read more
Viewpoint: A New Look at the Hydrogen Wave Function, C.T.L. Smeenk, Physics 6, 58 (2013)
For a more in-depth look, see
Taking snapshots of atomic wave functions with a photoionization microscope. A.S. Stodolna. PhD thesis, Radboud Universiteit Nijmegen, 2014.
The first images of hydrogen s orbitals were obtained in 2013 by physicists in the Netherlands.