We introduce characteristic classes for the spectral sequence associated to a split short exact sequence of Hopf algebras. These classes can be seen as obstructions for the vanishing of differentials in the spectral sequence. We give a decomposition theorem and interpret our results in the settings of group and Lie algebra extensions. As applications, we derive several results concerning the collapse of the (Lyndon-)Hochschild-Serre spectral sequence and the order of characteristic classes. (C) 2011 Elsevier Inc. All rights reserved.