What are some examples of non-homologous sequences having similar functions?
Serine Protease Catalytic Triad
This is classic example of convergent evolution in catalytic mechanisms. Shown below are chymotrypsin (4CHA, green) and subtilisin (1ST2, blue) aligned on their catalytic triads (Ser/His/Asp, darker colours). There is no obvious sequence or structural similarity.

You can check out the MEROPS database for a list of protease clans (&quot;represents one or more families that show evidence of their evolutionary relationship&quot;) and this paper for a discussion on convergent evolution of the catalytic triad:

Although the biophysics of rate acceleration are intricate and sensitive to even minor structural perturbations, evolution has converged on a catalytic triad (or diad) with a reactive Ser, Cys, or Thr nucleophile more than 25 separate times to facilitate central biochemical reactions such as hydrolysis, transacylation, and phosphorylation.