Feature selection: Information leaking if done before CV-split?
If you do not use target for feature selection, then there is no leakage and you can apply this on the whole dataset. The methods that you mention: checking correlation between features and checking variance of each feature belong here.

If you do use target for feature selection, then you should use 'train' subset only. Otherwise there is a leakage. The examples here would be (i) you check correlation between each feature and target (ii) use build Random Forest model and take best features