The design of spatio-temporal power allocation schemes is considered for space-time coding over spatially correlated multiple-input multiple-output (MIMO) channels. The focus is on linear dispersion (LD) space-time codes that are constructed to maximize the mutual information between the input of the space-time encoder and the output of the channel. While perfect channel state information (CSI) is assumed at the receiver, three cases are considered for the CSI at the transmitter: 1) perfect CSI is available, 2) only statistical CSI is available, and 3) partial CSI in the form of a B-bit quantized channel information along with the statistical information is available. In all the three cases, it is shown that the optimal temporal power allocation is uniform. The optimal spatial power allocation for the case where only statistical CSI is available was studied previously in  where it was shown to be a nontrivial function of the spatial correlation. Here, the cases of perfect and partial CSI are studied. For the perfect CSI case, it is shown that it is optimal to excite only one spatial mode. For the partial and statistical CSI cases, the optimal allocation excites multiple modes, in general. However, it is attractive to use a low-complexity scheme that excites only the dominant spatial mode. We show that this low-complexity scheme is near-optimal in two settings: 1) large receive antenna asymptotics, and 2) for fixed antenna dimensions, when the transmit and receive covariance matrices are ill- and well-conditioned, respectively. Based on the optimal schemes for the extreme cases of perfect and statistical CSI, low-complexity spatial power allocation for the case of partial CSI is considered. Simulation results indicate that even in this case, exciting one spatial mode leads to a minimal loss in performance over the optimal spatial power allocation scheme.