I have recently learnt about different tuning methods, and 12-tone equal temperament is a very well-designed compromise. However, are there usage scenarios where it's better to use just intonation compared to 12-tone equal temperament? How can I attempt to do so? (My primary instrument is piano, if that's relevant.)
Just about any time instruments are in use which can (and do) play using JI. So, if instruments such as guitar, piano, organ are part of the ensemble, it's going to be 12tet as the order of the day.
Using instruments such as those from the violin family, trombones and obviously voices, all of which can play those notes which sound more in tune using JI will produce a more harmonious sound particularly together, compared with 12tet.
It seems that they will naturally stray to JI in their playing anyway, which isn't possible for those tuned to 12tet.
Actually all amateur bands with instruments that allow some degree of freedom in the tuning, as well as choirs, should use Just Intonation for harmonies.
It is relatively easy to learn to recognise a just tuned chord and to adapt towards just tuning. But its nearly impossible to adapt towards 12-TET. One problem in 12-TET is that the third is somewhat dissonant. Even worse: consonance increases in both directions from a third, on the one side there is the just third, on the other the pythagorean, which both are more consonant than 400ct. So in a larger ensemble, the major third should always be intonated a little bit narrower than in 12-TET. This allows adaption towards one pitch. Otherwise the players may be locked in their consonance optimum and have no easy way to meet in one exact pitch.
There are surprisingly many instruments that allow some realtime tuning. Guitars can bend up, brass instrument are easier to bend down, while bending up is possible to a certain extent. Even woodwind instruments such as recorders allow some flexibilty.
On the other hand: You should always consider playing a piece in the tuning it was composed for. Tuning is nothing about pitch or transposition, it defines the tone colour. For example there exist so called Hamiltonian cycles in the Tonnetz of thirds and fifth, which can be used to visualise major and minor triads. When you play them in the proper well-tempered tuning, the tension rises and drops according to the mistuning of the chords. This gives the phrase an overarching tension dynamics, that is lost in 12-TET (Yes, 12-TET is not really well-tempered).
Tuning clashes can easily produced if the melodies of different voices tell another JI-story than the final harmony. Most choirs find them, when they first come to the corresponding position in the piece. As far as I know there is nobody really knows, how a choir really intonates. It is somehow related to JI, but there are melodic compromises. The latter can be trained to a certain extend, so that pianists tend more towards 12-TET than other instrumentalists that are more dependent on just intonation such as strings (open strings on violins and guitars have an important impact on the sound).
You can try it out yourself: Record a piece in different tunings and listen to them later. The question is always which one has more life and which one has a more sterile atmosphere. So, analytical hearing is not very helpful in this question. And it depends always on the composition which tuning is the best. Some pieces that are made for 12-tet sound mistuned in any other temperament, while others start can give you exciting new insights with another special tuning. It is said (I don't have the reference at hand) that Bach used the wolf as a feature of the tuning in certain compositions.