Buckets of water in the cold vs ice trays in freezer

Why does a bucket of water left outside in the cold will most likely break (it happened to me; filled a bucket for cleaning intentions, then something happened and I forgot it outside - then the winter came and when I came to retrieve it, the bottom was cracked)

I am not sure how valid it is to say that it will "most likely break" just after some anecdotal evidence, but we can for sure say that in your case it did break.

Why doesn't the ice expand upwards, why is the bottom broken? Could the sides break as well?) And so, why does the phenomenon not occur with ice trays?

The expansion will occur in all directions, so the breakage will just occur wherever the weakest part is. Of course ice trays are designed to freeze water, so they are designed to not break. The plastic is sufficiently strong enough so that the expansion of the ice cubes will not break them. Additionally, many ice cube trays do not have rectilinear shape - they have slanting walls such that the forces of expansion can be relieved by lifting to accommodate the volume expansion.