The half whole diminished scale is one of the most commonly used scales in jazz soloing. Its main function is to add alterations to dominant 7 chords. This creates a great deal of tension which usually resolves when the dominant moves to the tonic chord. I have included a PDF which explains the alterations added to a dominant 7 chord, 2 lines which use the diminished scale and a sheet explaining patterns within the scale.
Once you have become comfortable with the half/whole diminished scale, you can then start looking for interesting patterns within it. If we look at the notes again, we will see that there are several major triads within the scale (Please notes that for to be clear, enharmonic equivalents are used): C–Db–Eb–E–F#–G–A–Bb
C Major Triad: C E G
Eb Major Triad: Eb G Bb
Gb Major Triad: Gb Bb Db
A Major Triad: A C# E
These 4 major triads ascend in minor 3rd intervals. You can use this to your advantage by playing them in sequence over a dominant 7th chord. This is a great sound with a lot of tension which has a great release if you resolve the line. Once you have tried this in all keys, try playing the triads out of sequence. For example, you could play a C major triad, then a Gb Major triad then a Eb major triad and then an A major triad. You do not have to play all 4 triads every time. Using just one or 2 also sounds great.
Make sure you practice exercises in all 12 keys and listen out for the scale when listening to jazz. You’ll start to hear it all over the place!