Susan has given you both the British English (BE) and American English (AE) usages. Susan has also pointed out that "over the weekend" (but not: "over weekends") is commonly used, at least in AE. I just want to say that we *can* use other prepositions, in certain contexts: >I will be busy over the weekend. We can also say this with other prepositions, for example: >I will be busy during the weekend. (common) >I will be busy through the weekend. (common) >I will be busy throughout the weekend. (Not as common.) >I will be busy for the weekend. (rare) Or Just >I will be busy this weekend. (No preposition.)

American usage: on the weekend, on weekends British usage: at the weekend, at weekends

If you are talking about weekends generally, use "on weekends" example: "I enjoy taking trips on weekends." Never use "at weekend". If you are talking about last weekend or this coming weekend, say "over the weekend". example: "I took a trip over the weekend." or: "I will take a trip over the weekend." Hope this helps.

I speak New Zealand English, which is very close to British English & I have never heard "at the weekend" used in this way.