Punk is a very distinctive sub genre of rock with it's own very interesting sound, look, and persona. Which band was the first band that was classified as a punk band and start this trend?
I would actually disagree that punk is a distinctive subgenre. Punk is a pretty broad variety of sounds unified by philosophy and some aesthetics.
Punk as a label for a music and culture is generally considered born in late 70's London. Some people would actually say that punk traces back as much to Malcolm McLaren, who created the Sex Pistols and was otherwise influential in the culture, as much as to any band. No doubt devotees of UK punk could name precursors.
But many would argue that Punk was born in early 70's New York, with groups like the Ramones, and arguably, even the Velvet Underground. I've heard Detroit's MC5 labeled as punk, and they started in the 60's. And some people would include the 60's garage rock scene as punk, which is how you get claims like in Johnny Bones' comment. The Wikipedia Timeline of Punk labels that period "protopunk."
Sorry, you'll rarely get a straight "who was the first" answer out of me (in any genre.)
To expand on the New York roots of punk, folk singer Jeffrey Lewis has a great little song about the origins of punk that goes all the way back to the 50's.
Check out Jeff's history of punk here: https://www.youtube.com/watch?v=fnmrAthgi8g
I would personally argue that most of what he is talking about in the song is what was going on in music that lead up to Punk (what Joe calls protopunk in the post above), he does however mention that in 1971 Lester Bangs was the first person to use the word "Punk" in print to describe enthusiastic teenage rock music, and in 1972 Lenny K coined the term "Punk Rock" in the liner notes of a Garage Rock compilation called 'Nuggets'.
As far as what we know now to be the spikey hair, torn clothes template look of punk rock as made famous by the Sex Pistols comes directly from musician Richard Hell. Malcolm Mclaren (the manager of the Sex Pistols) credits himself for taking Richard Hell's look back to England in 1974(ish) to feed to the London youth via his clothing shop "Sex", where his wife, fashion designer Vivienne Westwood recreated the Richard Hell look with the addition of bondage gear, tartan patterns and chrome studs that we recognize today as the quintessential punk look.
Punk covers such a broad range of sounds, styles, and subgenres that I don't really think this has a definitive answer.
Punk in New York, London and Australia all sort of happened independently around about the same time in the early 70's. I do believe the Damned were the first punk band to release a single amongst the 70's English bands, as were The Saints in Australia.
But then as mentioned above you had the proto punk bands of the 60's such as The Stooges and The MC5 who were definitely punk in sound and attitude.
The Lenny Kaye Nuggets also mentioned above, and I do believe it was ? and the Mysterians who did 96 Tears.
Go back to the fifties and you had the original rockers who all embodied the "Spirit of Punk" Jerry Lee Lewis, Eddie Cochran, Gene Vincent, early Elvis.
The blues of Robert Johnson?
There really is no definitive answer to What band was the first to be considered punk? Everyone's opinions will vary greatly.