Does anyone know of any airs that have corresponding reels (or jigs?) like “The Green Fields of America”? Thanks.
Humpty Dumpty-Off She Goes-Anything for John Joe 
Search at Mudcat for something like “Irish jig song.” Rather bottomless topic here. Leg of the Duck, Polly Put the Kettle On, Father O’Flynn, Pretty Molly Brannigan, Rakes of Kildare are a few that come to mind.
I know some rather offensive verses for the Dingle Reggata too but that’s probably not suitable for an…international audience.
Wow, man you guys are quick! Thanks. 
The Blackbird is played as an air and also a hornpipe (set dance?) and a reel.
Also Sean O’Duibhir an Ghleanna is played as an air and a set dance (there are at least 3 versions of the tune).