some salient quotes:
The common custom nowadays is to use permit all soaps, even when containing non-kosher materials . Although there are Gedolei Yisroel, past and present, that are known to have been privately cautious in this area, vigilance from soaps is considered a chumra. The accepted practice amongst the masses is to use all soaps.
A lenient position based on the gemara in Avodah Zarah should apply even if dishwashing soap would contain non-kosher components in significant proportions. However, it is interesting to note that in reality the likelihood that a soap would contain non-kosher or animal derivatives is very minimal .
Like all other issues, consumers should consult their Rabbonim for direction.