Does a (public) database of theorems exist, as integer sequences are cataloged in the Online Encyclopedia of Integer Sequences (OEIS)?
The main Metamath page describes the project, the Metamath language, and programs and databases available for use.  The Metamath proofs are mechanical, and may or may not be useful because of their tedium and lack of insight.
An alternative to Metamath is Ghilbert , which looks much nicer. For example, contrast the Ghilbert proof of Euclid's Theorem  with the Metamath proof of infinitely many primes.  Unfortunately, Ghilbert does not seem to have an indexed database of theorems like Metamath does.
Some ad hoc lists of theorems that do not include canonical identifiers are:
The following is a list of some related answers on math.SE. However, these answers did not address a canonical indexing of the theorems.
- The most common theorems taught in Abstract Algebra
- Does anybody know of a site that has a set of all theorems?
-  http://us.metamath.org/mpegif/mmset.html#overview
-  http://us.metamath.org/mpegif/mmtheorems.html (inline GIF images) and http://us.metamath.org/mpeuni/mmtheorems.html (unicode)
-  http://us.metamath.org/
-  http://ghilbert-app.appspot.com/
-  http://ghilbert-app.appspot.com/edit/peano/peano_thms.gh/euclidthm
-  http://us.metamath.org/mpegif/mmtheorems99.html#mm9884s
I might also add that ProofWiki is a community project to do such, complete with proofs.