In my table I have firstname and last name. Few names are upper case ( ABRAHAM ), few names are lower case (abraham), few names are character starting with ucword (Abraham).
You don't need regexp to search for names starting with a specific string or character.
SELECT * FROM `test_tbl` WHERE cus_name LIKE 'abc%' ;
% is wildcard char. The search is case insensitive unless you set the binary attribute for column cus_name or you use the binary operator
SELECT * FROM `test_tbl` WHERE BINARY cus_name LIKE 'abc%' ;