I get the impression that there are two classes of games, ones that have been solved and for whom a dominant strategy exists, and ones that have not been solved and for whom a myriad of strong but not necessarily dominant strategies exist.
Is it ever the case that a game has more than one non-equivalent dominant strategy? Or do all cases of games with multiple strong strategies belong to the set of games that have not been solved?
One cannot have two strictly dominant strategies $\sigma_i$ and $\sigma_i'$ because as $\sigma_i$ is dominant, for all opposing strategies tuples $\sigma_{-i}$: $$ u_i(\sigma_i, \sigma_{-i}) > u_i(\sigma_i', \sigma_{-i}) $$ but as $\sigma_i'$ is dominant, for all $\sigma_{-i}$: $$ u_i(\sigma_i', \sigma_{-i}) > u_i(\sigma_i, \sigma_{-i}) $$ which poses a contradiction. If you mean 'weakly dominant,' so long as your definition of weak dominance is standard (i.e. requires at least one strict comparison), the same argument holds.