The ones that move most are where almost poetic lyrics convey deeper stuff of life - and where the melody and harmonic structure are a perfect fir with that message. Most of those are older standards, but some more modern takes will do it as well. Seeing Alison Krauss above reminds me of one - "A ghost in this house". Clapton's "Tears in Heaven" can be moving when done right.
There are quote a few more - mostly older - some from great stage shows.
