Ooh, this sounds like fun!
SONGS:
Is it too obvious to suggest Rock Lobster (B-52s)?
Beast of Burden (Rolling Stones) - not sure how "monstrous" this would be, but maybe they stampede?
Lady in Black (Uriah Heep), a ghostly apparition that when she appears might bring weal, or woe, or neither...
White Rabbit (Jefferson Airplane), another randomly-appearing creature that always leaves a trail of chaos in its wake.
BANDS:
Glass Tiger seems pretty obvious - a hard-to-see tiger made of glass or crystal.
A Flock of Seagulls (for these purposes, A Flock of Seaghouls) - marine ghouls that travel in packs.
Talking Heads could be a hydra-like monster whose main abilities are confusion, charm, and enthrallment.
Whitesnake - too easy, some sort of warm-blooded reptile that lives in snowy regions.
Rolling Stones - self-animated rocks. Again, too easy.
Beatles - huge locomotive-size beetles that club you to death with their antennae.
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If more come to mind I'll post them later.