In my opinion, the best evidence that I can find is testimony from the girls' father, Dave Olsen. The testimony was received via telephone by Slate.com author, Angela Tchou:
"We were just told by the doctors" that the girls were fraternal, says their dad, Dave Olsen, when reached today by phone. (Can you tell Mary-Kate from Ashley).
It is possible that the doctors were wrong ("as many as twenty percent of all twin births are misidentified as fraternal or identical"), or that Dave misremembers or is for some other reason inaccurately portraying the situation.
Other purported evidence (1 inch height difference, handedness difference) is not dispositive. 20.7% of monozygotic twins had different handedness in this study (see Table 5). Height is affected by childhood illness and nutrition.