Red veins in my whites: Is this normal?

Lately, I’ve been noticing these tiny red veins popping up on my whites, mostly toward the inner corners near my nose. It makes my eyes look kind of cloudy...
I actually went to see an eye doctor a few months back; eye pressure was fine, retina looked good, and my prescription is still sitting at -2.50...
Where do these annoying little lines even come from? I don't want people thinking I'm constantly tipsy...

I’m no medical expert by any stretch of the imagination, but please, for the love of God, stay far away from those Visine eye drops. It might seem like they work at first because they constrict those blood vessels and make your eyes look clear, but it's a total trap. It’s just like using a rubber band that you keep stretching and snapping back; eventually, those vessels just lose all their elasticity, get completely worn out, and end up looking even more swollen and red than they did before you started.

Man, I've been using Visine for years! What if it wasn't because of that?