What are experimental techniques to probe band structure of solids? What are experimental techniques to probe band structure of solids? I am not looking for exhaustive list. Few simple to understand techniques will suffice.
 A: To actually measure the 3D band structure is quite a difficult proposition. In materials that are quasi-2D (materials that are weakly bonded along one axis), the best probe of band structure today is angle-resolved photoemission spectroscopy. However, it only probes the band structure of the occupied states. For instance see the Fermi surface map of Sr2RhO4 below.
Fermi Surface of Sr2RhO4 http://www2.ph.ed.ac.uk/~rperry1/rhodate%20arpes.jpg
Other than ARPES, inverse photoemission spectroscopy is used to probe the unoccupied states' band structure. Quantum oscillation experiments (Shubnikov-de Haas, de Haas-van Alphen, etc.) provide valuable information about the Fermi surface, but must be conducted under a large magnetic field, which may itself alter the Fermi surface in some cases. Scanning tunneling microscopy provides valuable information about the density of states of a materials band structure. Optical experiments can yield information about the size of the band gap in insulators. Also, DFT (density functional theory) calculations are routinely performed to estimate band structure in materials. Obviously, these probes can give you more information than the quick description here, but I will stop here for sake of brevity.
All in all, it must be said that there is no one probe that can provide the band structure of all materials effectively. Most of the time one deduces the characteristics of the band structure from a wealth of experiments and calculations (numerical and theoretical). Hope this helps! 
A: Ashcroft & Mermin devotes a (short) chapter (14 in my edition) to experimental techniques to measure Fermi surfaces (note they focus mostly on metals).  These include de Haas - van Alphen, magnetoacoustic measurements, etc. For semiconductor band structures there are various techniques with (angle resolved) photoelectron spectroscopy being one of the more common ones used these days. 
