1 The Step-Up
The most basic single-leg exercise that every lifter should master. Progress this movement by increasing the height of the step or by adding load. Make sure to stand up and come to full extension at the top of each rep before making contact with the non-working leg.
2 The Lunge
Another fundamental exercise. Forward lunges will hit the quads more, but can often lead to knee pain in the patellar/quad tendons if overdone. Walking lunges offer a better combo of quads, hamstrings, and glutes. Reverse lunges target the hamstrings and glutes the best and stress the knees the least.
3 Rear-Foot Elevated Split Squat
Also known as the Bulgarian split squat, this is arguably the top single-leg exercise. Elevating the rear leg places a much greater demand on the front working leg, improving strength, balance, and coordination. It does a tremendous job of strengthening the entire leg (quads, hamstrings, glutes) and is easily one of the most "value for time" exercises for improving single-leg strength.
4 Single-Leg Pistol Squat
The most challenging single-leg exercise. If you can competently perform a full range of motion pistol squat, you possess excellent balance and single-leg strength.
A great way to progress these is to start on an elevated surface and increase the height as you get stronger. Performing pistol squats on the ground presents a greater challenge because it requires strength in the hip flexors and quads of the non-working leg. Practice this exercise with caution; it can lead to knee pain if done improperly.