I don't exactly know in terms of the rest of continental Europe but in terms of Britain slavery still very much existed, at least up until the Norman Invasion in 1066.

It is cited in the Doomsday Book that approximately 10% of the population of England were slaves. Relevant Information. After William came to the throne slavery in England went on the decline. The reason for this is that William the Conqueror made it against the law to own a slave in England forcing a fine paid to him from offenders (see point 9).

I would,however, hazard a guess that slavery very much didn't decline in mainland Europe as so many tribal societies in Gaul, Germania, Greece, Spain etc took captured foes as slaves, as the Romans did.

In taking these into account, one can safely say that slavery didn't go into decline after Rome's collapse as slavery was a component of life before and during the Roman occupation. This can be seen in Britain, as cited above. Further to this, there was nothing to say otherwise against the owning of slaves. The Western Roman Empire by its end was predominately Christian and, unfortunately, the Bible did not condemn slavery so the newly converted populaces wouldn't think twice about forcing a prisoner to work for them.

Hope this answers your question.