Wrap them in newspaper!
Unfortunately, most supermarkets nowadays not only chill produce to death but also sell so many items pre-packaged in plastic baskets. Mushrooms really should not be stored in plastic at all, let alone be sealed with cling film. At least that's what I was tought when we went mushroom hunting when I was little...
The other day, I spotted a wee note on a packet that suggested to remove the cling film and to cover the mushrooms with kitchen paper. That sort of worked but I've found that wrapping the mushrooms incl. the basket in 2-3 sheets of newspaper (from a broadsheet) works even better.
If the mushrooms are super fresh, they last for days in the fridge. In fact, we finished some last night (Thursday) that we'd bought last Friday without chopping bits off, pealing them or throwing any away.