"At/on (the) weekend(s)"

It really depends on the situation.

But if you're just asking about grammar, all of your sentences are right and in use except A, B and G.

These are the reasons:

1. "at" and "on" are both used. The former in British English and the latter in American.

2. Cambridge Dictionary recognizes "at weekends" but not "at the weekends."

3. It isn't always so but "the weekend" refers to a specific weekend while "(the) weekends" means every weekend.

With the last, you can determine which one is "better" depending on your context.

Sources:

Cambridge Dictionaries

Merriam-Webster Learner's Dictionary

Everybody is missing the main point. The usage of prepositions is idiomatic. So it varies depending on the speaker.

"At the weekend", "at a weekend" and "at weekends" are used in British English; "on the weekend", "on a weekend" and "on (the) weekends" in American English.

Generally speaking, words which refer to a period of time take in, like "in the morning", "in the month", "in the daytime" etc. Words which refer to an exact point of time take at, like "at 9 p.m.", "at dinner", "at Christmas", "at noon" and so on. Words which refer to a day or date take on, like "on Monday", "on 18th", "on Tuesday morning" etc.

So according to this rule the word "weekend" should be the object of "in". But it is not. We have never heard "in the weekend"!

So the answer is the usage of preposition is merely idiomatic.

The answer is F, which I'll explain in two parts:

The reason for on as opposed to at is that at would be used for a time with less length, such as "sorry to disturb you at dinner." For the most part, the delineation occurs at the period of a day, example: "What are we doing on Friday?" and "What are you doing at 5:00pm?"

Why you need "the", which is to say that answer b is not correct, is that "weekend" is ambiguous by itself. Example: "are you free on the weekend so we can get together?" means this coming weekend or the implied weekend in reference whereas "are you free on a weekend?" just means any old weekend.

I would use 'on' because a weekend is two days (or more). 'At' is more particular, for a smaller place or shorter time, whereas on/in are used for longer durations or larger spaces. "Let's eat at an Italian restaurant at 9pm" against "Let's eat in downtown on Friday".

Going by this logic, 'on' should be used.

'The' is imperative because weekend is a common noun, and to add specificity to it, we use the article the. 'The' denotes person(s) or thing(s) already mentioned, under discussion, implied, or otherwise presumed familiar to the listener or reader.

So you should use 'the' too.

Hence, from your choices, F is the correct answer.

D could make sense too, if you have been disturbing someone for many weekends. So your 'disturbee', for lack of a better word, would know that you acknowledge the fact that you disturb him on most, if not all, weekends.