I'm guessing you are given a piece of music that you look at?
Key signature, starting and ending notes, accidentals; look for those first.
Modes should be pretty obvious; given the key signature, if there is a scale passage (or a passage outlining a general scale) not beginning on the tonic, just find which other degree it is starting on.
Octatonic, Whole Tone, Pentatonic are also very specific examples; once you are familiar with them in each key, then they stick out like a sore thumb. I've never really seen a blatant Octatonic scale, except I vaguely remember seeing something kind of like an octatonic-ish thing in Chopin's G minor Ballade (right after the octave section, when it ascends). If they are giving you short passages where specific scales are added in, then an octatonic should be quite obvious (only scales with 8 tones...)
I don't really know what you are referring to with the Lydian/Mixolydian; I've heard of the synthetic Lydian scale which is like a melodic minor, but other than that, not too sure. If you are just referring to them as parts of the 7 modes that you know, then just remembering the names of which scale degrees should be enough.
Always write out the key in question; write in the closely related keys (it's relative major; the 4th and it's relative minor, and the 5th and it's relative minor); any scale melody not on the tonic would be a modal scale, and the pentatonic, whole tone and octatonic should be quiet obvious if you know the patterns of tones/semitones in them.