Piano. How would I incorperate lydian mixolydian dorian phrygian scales into a jazz piece and in which chords?

Lydian scales - play over Maj7 or Maj9th chords, Mixolydian - play over Dom7 chords, dorian - play over any minor chord, phrygian - could be played over a minor chord with a b9, it would be very "outside" sounding.

Incorporating these scales into a solo - start by just playing the scales over the chord changes, then change it so you aren't starting the scales on the tonic each time, but maybe on the 2nd or 3rd scale degree or another.

Then when you have the mode in your head, pick out melodies from within those scales so you can form a melody that flows from one to the other.

You know, of course, that if you are playing a piece in the key of C, playing a D dorian, E phrygian, F lydian and G mixolydian, are all the exact same notes.

pentatonic scales, pentatonic scales, pentatonic scales. All you have to do is change the position or scale, that's it.

First of all you have to analyze the song's chords, then when you will have recognize the position of the chord on the tonality (ex. ii v i or i vi ii v, etc..) you can use the modes. Remember that Lydian and Ionian are different because the 4, phrigian works well also on dominant 7 (b9 or #9) chords. Try for exemple to play an E7 on your left hand and then play an E Phrygian on your right hand, it will sound "spanish". usually in a normal ii v i progression (ex. D-7 G7 Cmaj7) you will use D Dorian, G Mixolidian and C Ionian. All these scales have the same note because they all come from C major key but you have to pay attention on which note you will play on the downbeat, if you play a D dorian scale (8s) starting from D over a C maj7 chord you will have the notes D F A C on the downbeat which are not Cmaj7 chord notes so it will sound "out".