Recent experiments uncovered a mutational pathway between two proteins, along which a single mutation causes a switch in fold. Searching for such paths between real proteins remains, despite this achievement, a true challenge. Here, we analyze fold switching in the minimalistic hydrophobic/polar model on a square lattice. For this analysis, we generate a comprehensive sequence-structure database for chains of length ≤ 30, which exceeds previous work by five units. Single-mutation-induced fold switching turns out to be quite common in the model. The switches define a fold network, whose topology is roughly similar to what one would expect for a set of randomly connected nodes. In the combinatorially challenging search for fold switches between two proteins, a tempting strategy is to only consider paths containing the minimum number of mutations. Such a restricted search fails to correctly identify 40% of the single-mutation-linked fold pairs that we observe. The thermodynamic stability is correlated with mutational stability and is, on average, markedly reduced at the observed fold switches.