It's best to consider keys and modes separately.
In a sense, our major and minor keys are examples of just two very specific modes (Ionian and Aeolian-with-alterations, as I said).
You don't use the other modes "with" these two. The other modes are totally separate ways of making (composing) music.
However, in rock music, you do often get what seem like mixtures of the two ways of thinking (key or mode). Very few rock musicians know or care about the theoretical niceties, after all - they just go for sounds they like, learned from their influences.
You typically have major key sequences - to start with - but altered or coloured in some way that could be described (not entirely correctly) as modal. IOW, rock bands treat the major key as something that can freely be altered (usually by flattening some of its steps to make it bluesier, funkier or heavier), just as a minor key (traditionally) can have its 6th and 7th raised.
Take G'n'R's "Sweet Child o' Mine", a very typical rock song. The verse is "in D major" (tuned down half-step), in that the keynote is clearly D. But there is no A chord, and there's a C instead. (basic sequence: D-C-G-D)
So - this is not the "D major key", strictly speaking: because it has a b7 note and bVII chord. We could describe it as "D mixolydian mode".
However, the chorus does contain an A major chord, the usual V of D major. So that part is clearly "D major key".
Overall, I guess you could call it "mixolydian major" - major key but with the significant addition of a bVII chord (along with a normal V chord).
You can also describe it as "major key", but with a bVII chord "borrowed from the parallel minor" (because C major comes from the D minor key). "Modal interchange" is another term for this kind of "cross-pollination" of keys and modes.
IOW, there are a few ways a theorist might describe what these guys are doing - while the band themselves could hardly give a damn!
This is an extremely common combination in rock music - you hear it in a lot of Beatles songs too: mixolydian verse, major key chorus. In fact, mixolydian mode was one of Lennon's and Harrison's favourite sounds (tho they didn't know it had a name). McCartney preferred major and minor key sequences.
Eg "Hard Days Night" - G mixolydian verse (with classic G mix opening chord), G major key chorus and bridge
"Norwegian Wood" - E mixolydian verse, moving to E dorian, and lastly to an E major key (Ionian) ii-V. (Classic example of "modal interchange".)
"I Feel Fine" - mixolydian riffs, in the context of a bluesy G major key for the verse. The bridge is straight (non-bluesy) G major key.
"Day Tripper" - E mixolydian vamp and riff. (This tune, in fact, is an intriguing mix of dom7 chords, many not related to the E major key.)
"If I Needed Someone" - A mixolydian - bridge (unusually) in B minor.
"She Said She Said" - Bb mixolydian right through.
"Tomorrow Never Knows" - C mixolydian right through.
"Within You Without You" - C# mixolydian (classic standard sitar mode) almost right through (hint of dorian in the bridge).
The Beatles attraction to - and take on - mixolydian was a kind of mix of bluesy b7s and Indian drones. Reflecting their influences, of course.
The Stones also use mixolydian, but in a much more blues-based way that was to become staple in mainstream rock. Eg:
"Sympathy for the Devil" - v. similar to Sweet Child o Mine, in its D mixolydian verse (D-C-G-D) and D major key chorus.
"Satisfaction" - E mixolydian riff, verse and chorus; E major key pre-chorus ("I tried, and I tried...")
"The Last Time" - E mixolydian right through. Or - arguably - E mixolydian verse and A major key chorus.
In short, any time you get a major key with a bVII chord (eg D in key of E, or G in key of A) you have a kind of mixolydian vibe - esp if you don't have a major V chord.
Dorian mode is probably the next most common modal sound in rock (but not nearly as common as mixolydian, nor as common as normal major and minor key). From one angle, it's simply a minor version of mixolydian: mixolydian with a b3.
From another, it's a minor key scale, but with a major 6 degree - and with no raised 7th alteration (as in harmonic or melodic minor).
The archetypal example is "Oye Como Va" by Santana, which is totally A dorian. (Because it's a Cuban tune, and dorian mode is widespread in Cuban music.)
The verse of Van Morrison's "Moondance" is also classic A dorian mode, but it moves to the A minor key in the prechorus.
The other classic rock example is the solo section of the Doors "Light My Fire", in A dorian.
Pink Floyd's "Breathe" has a lengthy E dorian intro and verse, but then moves into a mix of E aeolian ("long you live") and E phrygian ("all you touch..")
Deep Purple's "Smoke on the Water" is a kind of bluesy G phrygian riff and verse, with a G dorian IV chord (C major) in the chorus. (IOW, they combine all kinds of minor scale sounds - phrygian, dorian, blues, aeolian, even a hint of locrian - without caring too much!

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The whole point here, remember, is to
identify what scales or modes a song consists of, by analysis. Modes are a system of
terminology to aid our description of what's going on music. (Although keys and modes are two different things, modal terms can sometimes help in describing the way rock musicians adapt major and minor keys.)
There is a widespread and mistaken idea that modes are some kind of alternative "moods" that you can "apply" to an existing tune when improvising. This is not the case.
A related confusion is to assume you need to think about the "relative" modes in a major key (the other modes of the same scale). Eg, to treat a Dm chord in the key of C as "D dorian". This is strictly correct in one sense, but is not musically useful, is a waste of time, and can be misleading.