There are many English folk songs and Celtic tunes and airs that use the Dorian Mode, Mixolydian Mode, and the Aeolian Mode (the Aeolian is perhaps not as exotic as the other two, so maybe these don't stick out as much).
Dorian tunes include The Swallowtail Jig, Road To Lisdoonvarna, Scarborough Fair, All Things Are Quite Silent (which also has a B Part that leans very heavily into Mixolydian), What Shall We Do With The Drunken Sailor, and many more. To modern ears, used to chord progressions, the signature chord sequence would be minor i - IV, and as with all of these modes, the bVII subtonic is used constantly to shift away from I, and to give a kind of cadence back home to I. (this is similar to how V7 cadences to I in tonal harmony, but it is a weaker tendency, and these modal progressions sound "folky" - the minor v is very much related to bVII, but in my decades-long struggle to understand pure diatonic harmony, I almost always use bVII, and throw in the minor v as a color chord here and there)
Mixolydian tunes include: As She Moved Through The Fair, Campbell's Farewell To Redgap, B Part to Old French, The Red-haired Boy, Susan O'Dwyer ( here is great link to the progression and audio for Susan O'Dwyer http://songsinirish.com/p/siun-ni-dhuibhir-lyrics-chords.html )
A signature Mixolydian progression would be I - bVII and this is found in tons of Celtic tunes, and in quite a few Grateful Dead jams as well. For the latter, I've always used I-bVII-IV-I - I almost don't ever get sick of that!!
Aeolian tunes are always harder for me to think of, but Black Is The Color and Poor Wayfaring Stranger are two that use the minor iv - which to my ear sounds just incredibly sad!!! (yes I know that's very subjective, but in our western cultures it seems to be somewhat agreed). The first four chords to Star of the County Down are one of my favorite Aeolian progressions:
Bm - G - D - A or i - bVI - III - bVII
I've seen some discussions about Star of the County Down having some dorian tendencies and movements, but the structural bones seem to be Aeolian, unless I'm missing something....