I like this one because you don't have to muddle anything, I just don't like bits of lime and mint floating around in my drink.
Start with a few icecubes in your glass. This serves as a buffer for the mint leaves, so you don't suck those up when drinking. On top of those few icecubes, tear up about 8 leaves of mint, I tear them into 2-4 pieces. Rub the rim of the glass with some mint, it'll make your drink smell great.
1 oz freshly squeezed lime juice
1 oz simple syrup
2 oz white rum
Fill up your glass with more icecubes (I like to roughly pound them so you have something between cubes and crushed). Cover the glass with a shaker and shake well. Top it off with a splash of club soda. Not a lot, just to give the drink a little fizz. Don't stir it in, people stir the drink with their straw themselves.