I know that the construct was originally used for scheduling, and in recent years has been found to be useful in areas such a Cryptography, Error-correcting code and Affine and Projective Planes. [Latin Squares and Their Applications, Jason Tang]
A mathematician friend at the Clay institute told me back in the mid '00's that research into the construct was exploding.
Are there other areas of applied mathematics where Latin squares are being utilized or researched?