Myth #1: Touching your toes before a run prevents injuries. There's very little evidence static stretches like that actually work. You should do dynamic stretches instead where you move around. Then do static stretches AFTER your workout when you're loose.
Myth #2: You burn more fat if you work out on an empty stomach. It might be true in theory. But it also means you have less energy for your workout, so it's kind of a wash. A study in 2014 had one group of women work out on an empty stomach for one month, and another group drank a shake. After four weeks, they all lost the same amount of weight AND fat.
Myth #3: A protein shake after your workout will build more muscle. The research is mixed on that one. It works if you're doing really INTENSE weightlifting. But if you exercise like a normal person, you might just be adding more calories to your diet.
Myth #4: It's really easy to get dehydrated, so drink water even if you're not thirsty. In reality, thirst is usually a pretty good indicator. And unless it's really hot out or you got DRUNK the night before, it's not that easy to get severely dehydrated.
You can also drink too MUCH water and end up with hyponatremia, where your sodium levels get too low.