This mostly applies to the Western empire, but it would be nice to hear about the process by which the Eastern legions were phased out. Mostly, though, I'm curious as to what we know about their final days. Did they just sort of peter out, settling down and joining the local population when they stopped getting orders? Or were there even any left by the time the empire in the west finally crumbled permanently? Was there any attempt to maintain discipline at that point? Did any of them support the new 'barbarian' rulers of their regions?
I don't have access to the details of the history as my book is at home, but the very last legions were the ones in Greece and Anatolia near Constantinople that weren't committed to and lost during the battle of Yarmuk against the Arab Muslim army. After that, Heraclius gave the order for the remaining legions to remain in defensive posture to conserve what was left.
These remaining legions, under not entirely clear circumstances because of the lack of surviving documentation, eventually transformed into the early Byzantine thema armies.
Source: Byzantium in the Seventh Century by John Haldon.
EDITED: To provide direct citation.
However you should take note, that there is evidence of soldiers carrying on "legionary tradition" of their fathers and grandfathers who had been soldiers in the empire, more than 2 generations after the collapse of the western empire.
"Now other Roman soldiers [in addition to the Armoricans], also, had been stationed at the frontiers of Gaul to serve as guards. And these soldiers, having no means of returning to Rome, and at the same time being unwilling to yield to their enemy who were Arians [the Visigoths presumably], gave themselves, together with their military standards and the land which they had long been guarding for the Romans, to the [Armoricans] and Germans; and they handed down to their offspring all the customs of their fathers, which were thus preserved, and this people has held them in sufficient reverence to guard them even up to my time [mid-sixth century]. For even at the present day they are clearly recognized as belonging to the legions to which they were assigned when they served in ancient times."
The "classic" Roman legions disappeared during the reforms of Diocletian and Constantine. Even after them there were military units called "legions" but they were very different than the classic ones in size, equipment and commanding structure.
By the time of the events we today call "the Fall of the Western Empire", even these "new" legions were a thing of the past. Roman army led by Orestes and Odoacer in 476 AD, consisted of Germanic foederati, and it was them who eventually abolished the office of the Emperor of the West. They continued serving their own leader Odoacer, who became a Roman patrician and nominally recognized the Emperor of the East.
The last 'Roman' official was the Magister militum per Gallias named Syagrius who controlled a chunk of northern France after the deposition of Romulus Agustulus. He was defeated by Clovis and the surviving Roman army integrated into the Frankish army.
So whether you'd call these forces a legion is up for debate but they were the last remaining units of a Roman state in the west.