In the Asterix comic books, it seems that just about anything could get you thrown to the lions, in particular incompetence/lazyness/dereliction of duty. But what kind of crimes would really have carried this penalty in the 1st century BC? And would Roman citizens, such as the soldiers that are usually threatened with this, have had to face the lions, or only outsiders?
Damnatio ad bestias, being thrown to the beasts was kind of a intermission entertainment reserved for condemned criminals before the main act - the gladiators. There was a class division in this in that only commoners (and the lowest of that) could suffer such punishment, the elite condemned to death would be strangled.
As for soldiers, it's possible for some crimes. Senior magistrates with imperium had the power to order executions and discipline was fairly harsh in the army. For things like falling asleep on watch they'd simply have the offender beaten to death by his mates for putting them all at risk. It seems those who fled from battle (I'm presuming Goldsworthy meant individuals who fled before a unit rout) could face crucifixion or damnatio ad bestias, both normally reserved for the lowest of the low. I imagine that the latter would be at the nearest ampitheatre in the nearest city - I can't imagine a commander would spare the guards to escort such a man all the way to Rome. We have a record that in 167 BCE Paullus had foreign deserters from his army crushed by elephants - but this was in part a political act.
As a side note, it seems for those putting on private munera, one could 'purchase' condemned criminals for damnatio ad bestias for 600 HS around 177-180 CE.
There is actually a narrative account of a damnatio ad bestias in the text of the martyrdoms of the Christian Perpetua and her companions at Carthage in 203. (The first part of the narrative, recounting the arrest and imprisonment of the upper-class Perpetua and her slave companions for refusing to sacrifice to the emperor Septimius Severus, is arguably the first surviving writing by a Christian woman, Perpetua, though the case is not airtight.) An interesting detail of their execution is that they seem to have been costumed and staked to theatrical sets so that their execution would mimic the storyline of some myth, which was a feature of some damnationes ad bestias. If I recall, Cicero and some other Roman notables mention the practice.
The Zliten mosaics from Leptis Magna in Libya date to c. 200. They depict several scenes of people condemned to the beasts and gladiators fighting beasts. Pretty graphic floor decoration. A typical day at the amphitheater featured gladiators (bestiarii) fighting exotic beasts in the morning (the venationes), the execution of criminals at midday, and traditional gladiator fights in the afternoon.
EDIT: I should note that the Zliten mosaics depict all three stages of a day at the amphitheater, including musicians playing a horn and a water organ, which should remind us that the games featured a "soundtrack" of sorts.
Source: I’m recalling some of this from Carlin Barton’s The Sorrows of the Ancient Romans: The Gladiator and the Monster. The rest is off the top of my head.