Ultracold atoms loaded on optical lattices can provide unprecedented experimental systems for the quantum simulations and manipulations of many quantum phases and quantum phase transitions between these phases. However, so far, how to detect these quantum phases and phase transitions effectively remains an outstanding challenge. Here, we show that Bragg spectroscopy of ultra-cold atoms loaded on optical lattices can be used to detect the ground states, the elementary excitation spectra and the corresponding spectral weights of many quantum phases. These phases include not only the well known superfluid and Mott insulating phases, but also other important phases such as various kinds of density waves, valence bond solids and supersolids and quantum phase transitions between these phases in optical lattices.