Well, first rule – you should not do this. But if there is good reason, consider using such query for searching in index-based arrays:

SELECT * FROM table WHERE your_field_here REGEXP '.*;s:[0-9]+:"your_value_here".*'

In case you have assoc array serialized you can use:

SELECT * FROM table WHERE your_field_here REGEXP '.*"array_key_here";s:[0-9]+:"your_value_here".*'

Of course it won’t be very fast but in small tables should be enough