Does grep offer a way to count the total number of matches it makes? The -c option only returns the number of lines that matched the regex, but in this case I have multiple matches per line.
There is a -o flag which indicates that only the matched subsection of the line should get printed.
Use that in conjunction with wc -l:
grep -o "part of line" | wc -l
man grep explains it as well.
As an alternative to the other answers, using just grep:
grep -o "seach pattern" somefile.txt | grep -c ""
The -o in the first grep outputs each match, and just the match - not the entire line (unless the entire line IS the match, of course). The -c in the second grep then counts 'em.
It's a few more characters to type (like, 4 or 5), but I find it easier to remember.