Sturdiness has little to do with it. It's strictly a matter of weight. While you can get single trailer axles that can handle as much as 8000 lbs, most trailer manufacturers try to stay around 5000 lbs/axle and use multiple axles if the trailer weighs more than that. The primary reason is tire size and cost - the tires needed to handle more than around 2500 lb each will be larger sizes with higher load capacities and can get expensive. So, if the trailer is light enough, a single axle is fine.
When looking at trailers, check the axle load rating and also the tire max load rating. The trailer will have a federally-required sticker on the left front that gives the axle GAWR and total trailer GVWR, and each tire has its max load embossed on the sidewall. Ideally the axle(s) will be able to carry the GVWR with room to spare, and the tires will exceed the max axle rating too. But trailer manufacturers tend to go cheap in that are and you don't often find that. If the trailer fender wells have room, you can add larger tires later and that helps both reliability and tire life.