Hi,
'I usually go shopping during the weekend' is commonly used by Singaporeans. However, the dictionaries I referred to state 'at the weekend' is BrE and 'on the weekend' is AmE.
Is 'during the weekend' correct? Yes, but with this comment. 'During' makes the listener focus on the fact that the weekend is a period of time. The result is that it sounds to me like you are saying 'The weekend is a long time and I usually do many things and one of the things that I usually do at some point on Saturday or Sunday is go shopping'.
Best wishes, Clive