I've noticed that almost every fan drawing of elves (as well as their portrayals in the movies) show them with long hair, both the men and the women. But did Tolkien ever write anything that proved that male elves had indeed long hair? I had always imagined them with short hair (the men, not the women).
There are a few passages where JRR provides explicit statements. Don't know that you can extrapolate to conclude that all male elves generally had long hair but JRR certainly indicated some of them did.
Elwe himself had long and beautiful hair of silver hue, but this does not seem to have been a common feature of the Sindar, though it was found among them occasionally especially in the nearer or remoter kin of Elwe (as in the case of Cirdan). – J.R.R. Tolkien, The War of the Jewels, “Quendi and Eldar” Elwe - one of the Nelyar, ancient Elves of Cuiviénen.
Then Celegorm arose amid the throng (p. 169). In Quenta Silmarillion this is followed by 'golden was his long hair'. In the Lay at this point (line 1844) Celegorm has 'gleaming hair'; his Old English name was Cynegrim Faegerfeax ('Fair-hair'), IV. 213. The phrase was removed in The Silmarillion text on account of the dark hair of the Noldorin princes other than in 'the golden house of Finarfin' (see I. 44); but he remains 'Celegorm the fair' in The Silmarillion p. 60. – J.R.R. Tolkien, The Lost Road and Other Writings: the Commentary to “On Beren and Tinúviel”
Fingon - He wore his long dark hair in great plaits braided with gold. – J.R.R. Tolkien, The Peoples of the Middle-earth: „The Shibboleth of Fëanor” . Fingon was a High King of the Noldor.
The hair of Olwë was long and white, and his eyes were blue– J.R.R. Tolkien, The War of the Jewels: “The Grey Annals” . Olwe was the younger brother of Elwe.
Tolkien drew an elf. That elf has long hair.
The only definitive statement I'm aware of regarding Elvish long hair is made in the latter Quenta Silmarillion phase 1 texts given in History of Middle-earth 10, regarding the Teleri:
But most it was their wont to sail in their swift ships upon the waters of the Bay of Elvenhome, or to walk in the waves upon the shore with their long hair gleaming like foam in the light beyond the hill.
Here "long hair" was an amendment from just "hair" in the 1937 Silmarillion. In the published Silmarillion Christopher Tolkien removed the "long", thus reverting it to the 1937 reading, for reasons I am not aware of (he doesn't note any such authorial change in the phase 2 revisions).
By this we can understand that the Teleri of Aman collectively had long hair. Most of the Elves we meet in Lord of the Rings (with some exceptions: Gildor, Glorfindel, Galadriel) are of the Teleri, but there's nothing I'm aware of to suggest whether or not the Teleri of Middle-earth also collectively had long hair, so you can just run with your own preferred image of them.
They had long hair
Finwë had long hair.
Ingwë had curling golden hair. Finwë (and Míriel) had long dark hair, so had Fëanor and all the Noldor, save by intermarriage which did not often take place between clans, except among the chieftains, and then only after settlement in Aman. Only Finwë’s second son by Indis had fair hair, and this remained generally characteristic of his descendants, notably Finrod. Elwë and Olwë had very pale hair, almost white. Melian was dark, and so was Lúthien.
The Nature of Middle-earth - "Hair"
Finwë's hair was no longer than other elves.
All the Eldar had beautiful hair (and were especially attracted by hair of exceptional loveliness), but the Noldor were not specially remarkable in this respect, and there is no reference to Finwe as having had hair of exceptional length, abundance, or beauty beyond the measure of his people.
The Peoples of Middle-earth - "The Shibboleth of Feanor"
Thus, all elves must have had long hair.