Hero morph suits are the (zentai-themed) costumes equipped by your residents to transform them into super heroes. Though you could send your residents without hero morph suits to fight in battles, having them wear one can really make a difference in their performance. These hero morph suits are not easy to avail at all, so an easier way to have suits for them is by earning Enemy Morph Suits instead(from repeatedly fighting the same types of enemies), and equip them just alike .
Although it's easier to earn enemy morph suits in the game, nothing beats having your residents equip those cool looking zentai-costumes. To earn them, you'll either have to purchase them with 500 Gold Coins in the lottery, or alternatively use 100 Friend Points. Both of which are not easy to come by.
There are also 5 different colors per type of suit and perform somewhat different from each other with regards to their skills, default stats, per level-up stat increase, and weaknesses (check "Hero / Characters" in the Beginner's Guide section to see the RPS mechanic of the game).
Check the charts below to see all available hero morph suits in the game, sorted from classes F-A, and S.