Peter TYou asked for some reasonably well-known examples. I've been searching the ABCs I've downloaded and come up with these for starters:
Dorian: Garry Owen, Cuckoos Nest, 17 Come Sunday
Mixolydian: Paddy O'Rafferty, Three Sheepskins, Jolly Beggarman, Morisco
Phrygian: White Cockade, Campbells are Coming, Bessie Bell & Mary Gray
Lydian: I found half-a-dozen tunes, but none that I'd heard of.
Locrian: Couldn't find any.
Let me know if you want any of these posting.
Wassail! V