Queries using POSIX regex

-- Case insensitive match
SELECT * FROM mytable WHERE columnName ~* 'this|that';

-- Case sensitive match
SELECT * FROM mytable WHERE columnName ~ 'this|that';

Queries Using ANY

SELECT * FROM mytable WHERE columnName like any (array['%this%', '%that%']);

Queries Using SIMILAR TO

-- This is translated to a regex query internally, so not effectively different than constructing the regex query yourself
SELECT * FROM mytable WHERE columnName SIMILAR TO '%(this|that)%';