Opposed stat checks are pretty great

Stat checks. Easy. 

Roll d20 and get equal to or less than your stat to succeed, rolling exactly your score is a major success.
Bonuses and penalties will be made up on the spot at the time. 

Pretty standard except that rolling on your score is a critical success which makes you succeed extra good.
Some people like grades of success but I don't really care for them on a regular skill check. 
Major success is fine though because I can go "hey you succeeded and it was way cooler than you thought!".

Let's step it up a notch with opposed stat checks.

Both parties roll d20 and try to get as high as possible while remaining under their stat.
Both under stat: Highest roll wins.
Both over stat: Lowest roll wins.
One under, one over: Under-roller gets major success (ie. Gets what they wanted plus extra good stuff!)
Rolling exactly your stat is a crit and a major success. If both sides crit, highest wins.

I'm sure I saw something similar a million years ago on some shithead's blog but I can't find it.
Anyway, I figure that unlike a lone stat check, someone is always going to succeed at a contest one way or the other.
Easy to understand and seems to be working great in play so far.