The physical mechanisms and ensuing material modification associated with laser-induced damage in multilayer dielectric high reflectors is investigated for pulses between 0.6 and 100 ps. We explore low-loss multilayer dielectric SiO2/HfO2 mirrors which are commonly employed in petawatt-class laser systems. The spatial features of damage sites are precisely characterized, enabling the direct correlation of the observed damage morphology to the location of energy deposition and the corresponding standing-wave electric-field intensities within the layer structure. The results suggest that there are three discrete damage-initiation morphologies arising from distinctly different mechanisms: the first prevailing at laser pulse lengths shorter than about 2.3 ps, while the other two are observed for longer pulses. Modeling of the thermomechanical response of the material to localized laser-energy deposition was performed for each type of damage morphology to better understand the underlying mechanisms of energy deposition and subsequent material response.