"And naked to the hangman's noose/The morning clocks will ring/A neck God made for other use/Than strangling in a string" - AE Housman
Telemachus (Homer, Odyssey, bk22 vvs462) hangs the collaborationist slave-girls, to deny them "an honourable death."
There is little sign of hanging in classical Greece. It is not specified in Plato's lists of actual (Republic, para361C) or suggested (Laws, bks5-12) punishments.
Athenaeus (Learned Men at Dinner, bk4 para155e) describes a Thracian game of pretend self-suspensions during booze-ups, which sometimes ended fatally.
Aulus Gellius (Attic Nights, bk15 ch10), quoting Plutarch, describes the mania among Milesian girls for hanging themselves, curbed when the authorities decreed their bodies be buried naked along with the ropes - "a grave disgrace".
In Greek Tragedy, Jocasta and Phaedra hang themselves, tainted heroines both. Likewise, Amata in Virgil's Aeneid (bkl2 v603), "an unseemly death", one denied burial by Roman pontifical law. A lapidary text (Inscriptiones Latinae Selectae, no7846) excludes such suicides from cemeteries. Legal eagle Ulpian (On the Edict, ch6l) observes: "It is not usual to mourn those who hang themselves".
Roman law codes (e.g. Digest, bk48 ch19 para 28 pref.) specify the Furca along with burning alive and decapitation as prime punishment for criminals, plebeians, and slaves. Isidore of Seville (Origines, bk5 ch27 para34) defines this as hanging or strangling the victim via their pronged head - forking hell!
For religious reasons, after Constantine the Great, this replaced crucifixion in late Rome and Byzantium, though traitors were sometimes strung up (Theophanes, Chronicle, ch184 pp4-6, Bonn ed.). Christian ideology (e.g. Augustine, City of God, bk1 chsl7-27) also reduced the suicide rate. A rare exception was the bankrupt scribe Melites who (1303) hanged himself (Pachymeres, Reigns of Michael and Andronicus, bk2 pps385-8, Bonn ed.).
Eusebius (Church History, bk5 ch16 paral2) asserts that Judas Iscariot and the heretics Montanus and Maximilla must have been divinely driven mad to hang themselves. Execution by the rope is little heard of in Rome, being (e.g.) unmentioned in the earliest law codes (The Twelve Tables), though Cicero (Against Verres, bk3 ch26 para66) mentions Sicilian victims "swinging from the trees". Garroting is more attested (e.g. Sallust, Catiline, ch55), albeit Tacitus (Annals, bkl4 ch48 paral) says it was obsolete by Nero's time.
"Go hang yourself" is a common expression in Plautus's comedies. "Choose your tree to hang from" was proverbial (Pliny, Natural History, Pref 29; Seneca, On Anger, bkl3 ch15 para2). Plebeians hanged themselves (Pliny, bk36 ch24 para1107) to avoid working on King Tarquinius Priscus's sewers - Les Battersby may come to mind. But in a case-by-case register of suicides over 1,000 years (Y Grise, Le Suicide dans la Rome Antique, Bude, Paris, 1982), only half a dozen involve hanging.
Finally, two cases for Falco. On 15 May 392, emperor Valentinian II was found hanging in his room: some contemporary sources cry murder, others say suicide. From Egypt (AD 173) comes this police report (Oxyrhynchus Papyri, voll nol para5l): "Today I was ordered to inspect the dead hanged body of Hierax and offer my conclusions. I inspected the body in the presence of his servant and found it hanged with a noose. This is my report."
"There are data of strange suicides that I shall pass over" - Fort, Books, p653.
(reprinted with permission of the Author; blame any typically graphic transcription errors on dm)