I like trying to think outside the box when I see lists of answers that seem split between two aspects of the question, so here goes.
Some previous answers get hung up on beauty being primarily "physical" and write off other attributes as mere 'tie-breakers', while other answers go to the other extreme by taking any definition of 'beauty' and implying that differences in species/races/etc. will be too different to compare with said definition.
I think some sort of compromise on both of these concepts would work well in this type of competition.
First, while we humans certainly weight physical appearance heavily when determining "beauty", even among humans this is not the only criteria. I find it hard to believe any human, without total hearing loss from birth, would fail to acknowledge that music can be beautiful upon hearing a selection that appeals to their personal tastes. At the same time, blind humans will still have a concept of "beauty" that does not include what would traditionally be called 'appearance'. Humans also find beauty in skills and talents (beyond their use as tie-breakers, just ask a sports fan about a 'beautiful play' made by a player of their favorite team).
Next, since the definition of beauty will certainly vary among the participant races/species, so use that. Maybe have each participating world submit their single most important criteria (or a select few criteria, 2-5?) for "beauty" to be used as judging criteria in the competition. This ensures that all participants are evaluated (within the limits of any subjective competition like this) on even footing with all others.
So there could very well be a "physical appearance" portion of the competition, proposed by us humans, and we may even fare relatively well in that portion (who knows?), but how would we do in the part of the competition dedicated to making noises audible in the widest range of frequencies at the same time, without the aid of external instruments, proposed by the species that has no eyes, and who's primary form of "beauty" is their version of singing through their 4 mouths?
If a human were to actually WIN this competition, it would almost certainly NOT be the same human that would win a more traditional beauty pageant here on Earth, but instead one that is "good enough" by our own physical appearance standards, while also appealing to as many other varieties of "beauty" presented by the other races. Maybe an attractive(appearance category) astrophysicist(intelligence category) singer(talent category) that can hold their breath for a long time (category submitted by an amphibious race, and singing talent just happens to help with lung capacity) and was born with 6 fingers on their right hand (from a race with 6 arms and 8-12 fingers on each "hand" that defines beauty based on the total number of fingers)