We consider offline detection of a single changepoint in binary and count time-series. We compare exact tests based on the cumulative sum (CUSUM) and the likelihood ratio (LR) statistics, and a new proposal that combines exact two-sample conditional tests with multiplicity correction, against standard asymptotic tests based on the Brownian bridge approximation to the CUSUM statistic. We see empirically that the exact tests are much more powerful in situations where normal approximations driving asymptotic tests are not trustworthy: (i) small sample settings; (ii) sparse parametric settings; (iii) time-series with changepoint near the boundary. We also consider a multichannel version of the problem, where channels can have different changepoints. Controlling the False Discovery Rate (FDR), we simultaneously detect changes in multiple channels. This "local" approach is shown to be more advantageous than multivariate global testing approaches when the number of channels with changepoints is much smaller than the total number of channels. As a natural application, we consider network-valued time-series and use our approach with (a) edges as binary channels and (b) node-degrees or other local subgraph statistics as count channels. The local testing approach is seen to be much more informative than global network changepoint algorithms.