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
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