Does anyone know more examples of two nonisomorphic connected graphs with the same chromatic symmetric function? The only pair I know is the one in Stanley's paper on c.s.f.'s [http://math.mit.edu/~rstan/pubs/pubfiles/100.pdf; p.5 of the PDF file]. I would especially like to have an example in which at least one of the two graphs is triangle-free.
I don't think there are any other published examples. I think your best bet is to look at the literature on "chromatically equivalent graphs" (graphs with the same chromatic polynomial) and do your own computations to find examples. (I assume that you wrote some code to compute the chromatic symmetric function when you investigated trees.)
This is quite an old post of mine. Subsequently, Rosa Orellana and Geoffrey Scott gave an infinite family of pairs of unicyclic graphs with the same chromatic symmetric functions [Discrete Math. 320 (2014), 1–14]. That's the best answer to this question that I know of. Stanley's original question for trees is still (frustratingly) open, so far as I know.