How are unstable transition states determined?
Experimentally, and surprisingly,  there is only circumstantial evidence for most short lived species in any type of reaction, i.e. no one has spectral or structural evidence for a transition state, such as often portrayed in textbooks, other than for example NaI dissociation or atom-diatomic molecule reactions e.g. H+OH  etc. done in ultra-low vacuum. 

For longer lived intermediates (picoseconds to longer) short pulse laser pump-probe (flash photolysis) methods have to be used to obtain spectral signatures but give no structure, and this method only works for light triggered reactions which are rare. An alternative is stopped flow, but the time scale is slow, many microseconds so usually too slow for most intermediates, and again does not give structure. NMR is immensely slow also. Time resolved x-ray only works on crystals. So far light driven reactions in proteins have been studied down to a few tens of femtoseconds.  There is a huge and exciting field here to be explored.