Setting aside a discussion of changing your database design...
A combined index (an index on many or most of the columns referenced in your query) isn't going to be of help for your query, which has a bunch of
colN = 'foo' predicates. MySQL is not going use that index to satisfy your query. Even if it were to use the index, there would still be other columns in the underlying table that need to be checked on essentially every row, so MySQL is very likely just to visit all the data pages and not use an index at all. (If you happen to have a GROUP BY or ORDER BY in your query, MySQL might be able to use the index to optimize those operations, especially if it was a "covering" index that included EVERY column referenced by your query.
On the other hand, IF you had a separate, individual index on EVERY column (as a leading column in the index) that was checked with an
OR colN = 'foo' OR colN = 'bar', it is possible that MySQL would consider using an "index merge" plan for your query.
But it would have to be an index on EVERY column. If your query is checking even just ONE column that is not a leading column in ANY index, then MySQL would have no choice but to examine every row in the table. So having separate indexes on "many" columns will not help your query, because it's very likely that NONE of the indexes will be used.
Even if you did have a separate index for every single one of the boatload of columns being reference, it's likely that MySQL's estimate of the total number of rows being returned (combined from each index) is too large, and MySQL is likely to decide that an "index merge" is too expensive, and opt for a full table scan instead.
In summary, your only two choices for indexes to help your query (and neither of them is a really good choice) would be:
1) a "covering index" that has leading columns that can be used to satisfy a GROUP BY or an ORDER BY clause (avoiding a "Using filesort" operation"
2) separate, individual indexes on EVERY column (as a leading column) that is checked by an
OR colN = 'literal' predicate in your query
But again, neither of those is likely to be a good choice.