I've heard this topic discussed before.
Last night, Psycho (1960) was the first slasher film, according to the guy I was listening to.
I think Psycho was kinda the first 'modern' horror film, in that it was set in 20th century America - there were no vampires, castles and cobwebs. The monster could live next door. That's what people found really terrifying, I think.
The two kills in Psycho definitely involve slashing but to me the film is mainly about tension, psychological horror, not slashing. Also, not all of the slasher tropes are present - e.g. there's no final girl. I'd call Psycho a proto-slasher - it's what slasher films evolved from.
Halloween (1978) is an early slasher film. The killer uses a slashing weapon, the promiscuous teens get picked off, and the goody-goody Laurie Strode survives as the final girl.
John Carpenter also paid homage to Psycho in his film - Dr Sam Loomis was named after Marion Crane's boyfriend. They thought about casting Jamie Lee Curtis because she's the daughter of Janet Leigh.
IMO, I think that The Texas Chainsaw Massacre (1974) could possibly be considered the first slasher film, with discernable slasher tropes. Leatherface kills people and/or dismembers people with a chainsaw (a slashing weapon), and the blonde-haired Sally Hardesty is the final girl.
I've heard Peeping Tom (1960) mentioned but I haven't seen this film.
Black Christmas (1974) is sometimes mentioned. I own a dvd of this film. It's good but the killer doesn't use a slashing weapon, from what I can remember.
What do you think?