Exhaustive enumeration of sequences and folds is conducted for a simple lattice model of conformations, sequences, and energies. Examination of all foldable sequences and their nearest connected neighbors (sequences that differ by no more than a point mutation) illustrates the following: (i) There exist unusually large number of sequences that fold into a few structures (super-folds). The same observation was made experimentally and computationally using stochastic sampling and exhaustive enumeration of related models. (ii) There exist only a few large networks of connected sequences that are not restricted to one fold. These networks cover a significant fraction of fold spaces (super-networks). (iii) There exist barriers in sequence space that prevent foldable sequences of the same structure to “connect” through a series of single point mutations (super-barrier), even in the presence of the sequence connection between folds. While there is ample experimental evidence for the existence of super-folds, evidence for a super-network is just starting to emerge. The prediction of a sequence barrier is an intriguing characteristic of sequence space, suggesting that the overall sequence space may be disconnected. The implications and limitations of these observations for evolution of protein structures are discussed. Proteins 2012. © 2011 Wiley Periodicals, Inc.