Heat the oil in a large nonstick skillet over medium-high heat. Add the spinach and cook, stirring occasionally, until the spinach wilts, 3 to 5 minutes. Season with salt and pepper to taste. Add the eggs and cheese and cook, stirring frequently, until the eggs are set, about 2 minutes.
Stack the tortillas on a microwave-safe plate, uncovered, and heat in the microwave until warmed through, 30 to 45 seconds.
Assemble the wraps by placing a quarter of the egg mixture on each tortilla. Top with 1 to 2 tablespoons of salsa, wrap burrito style, slice in half, and serve.