I only check it on things that are likely to go bad before I consume them. We don't drink milk very quickly, so I do get the most recent one I can find. For other stuff it's mainly a reminder, but the nose knows. Cream that spoils in the fridge won't hurt you, so it's mainly a matter of taste. I've used ultra-pasteurized cream several months after the date. For sauces, typically, not ice cream, for example.
Part of the problem is that there are three types of dates. "Best by" is a recommendation to consume by a certain date, but the quality depends a lot on your storage conditions. There's no safety issue in consuming canned food that's a decade old, but boy will it probably be bland and mushy by then. (Actually, there's more evidence that the BPA can liner is unsafe from day one.)
The second type is "sell by." Milk has this, and I have read that for best taste you should drink within seven days of opening or the sell by date, whichever is sooner.
Finally, we've got "use by," which is the closest to a true expiration date. These I pay more attention to, because that bagged lettuce is can get really digusting even a couple days past. Ditto for chicken.