Green’s Dictionary of Slang

1. attractive, esp. when seen as ‘cheaply’ so.

1831		Fraser’s Mag. III 679/2: there is great danger of every thing becoming. forced and unnatural, and all other qualities sacrificed to a catchy, stage-like effect, both as regards subject, composition, and execution.
1899		A.C. Gunter M.S. Bradford Special 176: ‘Well, have your way with me,’ remarks her mistress, resignedly. ‘Only make me catchy.’ ‘Ciel! Madame shall be a fashion-plate.’.
1903		‘Hugh McHugh’ I Need The Money 63: A Dago bootblack and a Long Island Reub – a catchy pair, believe me!
1947		G. Kersh Prelude to a Certain Midnight Bk I Ch. i: Her name was Catchy. [...] It is a fact that she was extremely attractive in those days, although her beauty was of the commonplace sort.
2006		C. Carr Our Town 260: I might borrow the ‘catchy title’ they’d just given their new newsletter.

2. tending to take an undue advantage.

1859		Hotten Dict. of Modern Sl. etc. 17: catchy (similar formation to touchy), inclined to take an undue advantage.
1864, 1867, 1870		Hotten Sl. Dict.
1873		Sl. Dict.