A rich tycoon of crime (Mr. Blob) tries every means at his disposal to kill the Man of Steel in Superman Comics #29 (May 1944). What I found interesting is that while searching for Superman’s “Achilles Heel”, the villain comes across the only time Superman was ever threatened: that time Superman’s was affected by colliding stars way back in June 1943 (see Superman Goes Bad). That’s almost a year in the past – nice in-story continuity there DC!
The tycoon almost succeeds – hypnotizing Superman and convincing him to commit suicide – but Superman fails to kill himself and wakes up from the hypnotism, putting an end to the villain’s plot.
Casting Superman’s super senses as a vulnerability is a cool idea, one used in the 2013 film, Man of Steel. Though Kryptonite had been introduced into the radio serial, 1944 was still years before it was introduced in the comics, so finding some means for Superman to be vulnerable hadn’t become commonplace in Superman stories at this point.