Great question @fredrik.
Yes we can apply triads in both a harmonic context, and in a melodic context. Below I have included more explanation and the most relevant lessons, seminars, and PDF downloads.
Triads In A Harmonic Context (USTs)
You are correct that triads can have harmonic applications such as in Upper Structure Triads - mainly when applied over dominant chords but also applicable to major, minor, and -7b5 chords.
The following 4 links contain relevant info relating to dominant chord USTs:
I would recommend printing the Upper Structure Cheat Sheet and stick it close to your piano or on your piano for quick and easy reference:
When playing jazz standards, if we want to harmonise a melody note with an upper structure triad, we follow this process:
Find a dominant chord on a lead sheet or in any tune that we are playing
Identify the scale degree of the melody (very important step)
Match the melody scale degree to one of the UST chord tones in the far right column
This will tell us which triad will work (sometimes we have more than one triad option)
Invert the right hand triad so that we have the correct melody note on top of the voicing.
After some time following this process, you will remember which USTs work with which melody notes. For example, every time we have a dominant chord with the 13th in the melody, we can play either UST 2 or UST 6. Often one will work better than the other, and sometimes it is just personal taste as to which one to choose.
Now, when we first try to apply this to a chord, an Eb7 for example, it might take you a few minutes to visualise the F Triad (UST 2) or the C Triad (UST 6) but after you have done this many times over the same chord (Eb7 in this case), it becomes second nature for our right hand to gravitate towards those triad shapes.
Pro Tip: If you can memorise the UST formulas from the table below, that will make the process much quicker for you. To start with print out the table and stick it close to the piano.
Triads In A Melodic Context
Triads can also be applied in a melodic context as a means to derive melodies when improvising. This concept is presented in detail in the following lessons, seminars and PDF files:
And PDF Files:
Another application can be found here in Tuomo’s lesson on Triad Pairs In Improvisation