This page details bugs of Rockin' Kats.
Willy's downward diagonal punch attack uses one incorrect tile, right in the middle of his head, though it's hard to tell unless you pause the game at just the right time.
Strange things can happen when both Willy and Big Eagle die at the same time. Willy's sprite may glitch if he touches the spikes, and if his hearts are empty when Jill descends on the propeller platform, the game will get confused and send Big Eagle back in...except his tilemap is no longer in memory, so you'll see the status bar and ground plane swoop in instead. Fortunately, the game gets itself back on track and restarts the boss battle a few seconds later.
(NOTE: This only works on the first Big Eagle fight!)