glue the trim on adhesive silicon caulk. Hold it in place with tape until the adhesive sets up.

Your tile layer should have replaced the short pieces with pieces cut to eliminate the gaps ...

Thanks - the tile layer was a DIY job and it's hard to see from the picture, but the tie was not deep enough to meet the back of the wood, so the trim was also to hide the difference there too. I'm sure none of that just made sense though! I would not have been able to get a good looking grout line either way I guess.

Either use a different profile of trim (i.e., quarter round) and nail to the cabinet, or use construction adhesive (I like Power Grab) to glue it to either the tile or the cabinet.

Lazy's suggestion is a good one too. If the trim is lightweight, caulk should hold it in place well enough, and would then be easy to pull out down the road when you DIY again. :-)