I remember playing solos by simply learning the scales and randomly shredding. Although this may look and sound impressive to many, I don’t really think that it is too musical. This type of soloing is very common with guitarists since they can hold the fret positions in a key and quickly shred notes, escpecially on an electric guitar. Not as easy on the piano, however it can still be done once you are familiar with the scales.
Hearing other people do solos in usch way sounded impressive to myself at first but then it becomes rather tedious to listen too.
What I like to do these days is play an actual melody for a bar, then simply repeat that melody for 4 bars. The melody doesn’t need to be exactly the same for each bar, in fact it sounds much better and improvised if the melody is slightly altered each time, with slightly different notes and timing.
This is like an intro and warm up at the same time. Once you repeat the melody for a number of bars, you can easily branch of into more random and complex solos while at the same time basing it in that original melody. I like to end the solo with something similar to that original melody.
By doing solos this way, this sounds more structured. Both musicians and non-musician can appreciate it.
Check out the below video for an example