I'm not sure how much you're into today's popular music, but modes are used extensively in pop music today. Here are some examples with their chords, keys and an explanation:
Work - Rihanna: Based on the notes that we find in the song, we could categorize it as B major, but the chord B major is not the chord that brings the most resolution. I would argue that the starting chord (the ii chord) brings the most resolution to this song. Here are the chords:
C#m D#m Emaj7 F# (each chord is 2 beats each)
If you've ever heard of "trap" music, it's a sub-genre of EDM, but it's really today's sound for rap music. Trap music uses the phrygian mode and aeolian mode exclusively. The main issue with writing out chords for the "trap" sub-genre is that they don't really use chords explicitly, but they can be implicitly derived by analyzing melody notes and bass notes in the beats.
Hotline Bling - Drake: This would be considered F major, however, the chord that resolves this progression the most would be Bb major (the IV chord). Here's the chord progression:
Bb Am Bb Am (each chord is 4 beats each)
Royals - Lorde: If we're looking for a major key to represent this as, then it would be G major. However, we're really in a mode because of the chord progression.
D D C G (each chord is 4 beats each)
In the key of G major, we'd have V, V, IV, I since D is the V chord, C is the IV chord and G is the I chord. When a phrase begins on something other than the I chord, this usually indicates that we're in a mode. In the case of this song, the D major chord brings resolution and the C and G major chords create tension.
This One's For You - David Guetta ft Zarra Larsson: In major key terms, this would be in C major, but the chord that resolves at the end of the song is Am, so we're in the Aeolian mode. We wouldn't call it A minor (usually in the harmonic form) because there's no G# anywhere in the song. Here are the chords:
F Am C F (each chord is 4 beats each)
In jazz, modes are used for soloing all the time.
In the following jazz standard, "Tenderly", we'll take a look at how to use modes in soloing. The key of the song is Eb major.
Note: For each chord, there may be more notes played than just the typical notes for the chord listed. Colour tones (9s, 11s and 13s) are frequently omitted from scores for simplicity and it's also assumed that the player will know how to add them.
Bar 1: The chord is Ebmaj7(Eb G Bb D) and the most typical method to choose notes for soloing would be to use the Ionian mode (Eb F G Ab Bb C D...). For the I and IV chord of a key, we can use any of the major modes for soloing, which means that we can also use Eb phyrgian (Eb F G A Bb C D...). Both modes fit with the notes of Ebmaj7.
Bar 4: The Ab13 chord indicates that we have the notes Ab C Gb Bb and F, most typically, but there are other variations. For soloing, since this is a dominant 7th chord, one mode to use would be Ab mixolydian, which is Ab Bb C Db Eb F Gb...
Modes By "Brightness" (For Soloing)
I'll list the modes from light to dark below
Lydian: A major mode (#4 is the distinguishing note)
Ionian: A major mode (7 is the distinguishing note)
Mixolydian: A major mode (b7 is the distinguishing note)
Dorian: A minor mode (6 is the distinguishing note)
Aeolian: A minor mode (b6 is the distinguishing note)
Phrygian: A minor mode (b2 is the distinguishing note)
Locrian: A minor mode (b5 is the distinguishing note)
There's a great channel on YouTube here (Rick Beato). Hope this helps.