For me, the answer to using headphones or earbuds on the bike is an emphatic 'no' for safety reasons.
Situational awareness is critical on a bike, and headphones impact mine too severely for comfort. If you disagree, more power to you . My personal answer is no.
Happily, there's an alternative! They are called O-tus helmet speakers, and they look much like earbuds. Unlike earbuds or headphones, though, the O-tus speakers mount to your helmet a few inches above and ahead of your ears.
Because they aren't plugged into the ears, I've found I am comfortable using these speakers on the road or MUP. I hear my music without losing my awareness of or ability to hear non-musical goings-on around me.
Full disclosure: I won my O-tus speaks through a contest here on TwoSpoke. I didn't expect them to work as well as they have, especially for cold-weather cycling. I plan to buy a set for Mrs. Newleaf when the weather warms up a bit.