Although I can't find a definite reference in the MySQL documentation, due to the index structure, regex matches can not use index lookups.
They may scan indexes using the regex instead of scanning the full table, but any regex search will be more or less equivalent to a full table scan.
If your regex starts with a constant string, you may want to structure the query as;
SELECT * FROM TableA WHERE field LIKE 'tr%' AND field RLIKE 'tr[uae].*'
to get some index help from the LIKE that - if matched from the start of the string - can use the index to its advantage.
EDIT: Since your selects are fairly simple and match from the start of the string, you can just use LIKE to make them work, which will utilize indexes well;
SELECT * FROM TableA WHERE field1='2012' OR field1 LIKE '2012/%'
SELECT * FROM TableA WHERE field1='2012/10' OR field1 LIKE '2012/10/%'
SELECT * FROM TableA WHERE field1='2012/10/3'
The reason to split the comparison into two is to avoid a search for 2012/1 that would match 2012/10 with just a LIKE that covers both cases (ie LIKE '2012/1%').
2012isagoodyearand need to reject it?