If you're working away from the primes dividing the level, your curves have semi-stable reduction, and have canonical log-smooth log structures. For any pair (X,D), where X is smooth and D is a divisor with normal crossings, there is a log structure given by the set of functions in O_x that are invertible away from D. In your case, I think you take X to be the universal curve, and D to be the divisor at infinity. Forgetting a coprime level structure yields a map with vanishing log-cotangent complex.

References (may not have your precise statement):

• F. Kato. Log smooth deformation theory
• M. Olsson "Universal log structures on semistable varieties"

Olsson has some other papers that might be useful. He takes them off his web page when they get published, but sometimes you can find them with Google Scholar.

If you're working away from the primes dividing the level, your curves have semi-stable reduction, and have canonical log-smooth log structures. For any pair (X,D), where X is smooth and D is a divisor with normal crossings, there is a log structure given by the set of functions in O_x that are invertible away from D. In your case, I think you take X to be the universal curve, and D to be the divisor at infinity. Forgetting a coprime level structure yields a map with vanishing log-cotangent complex.

References (may not have your precise statement):

• F. Kato. Log smooth deformation theory
• M. Olsson "Universal log structures on semistable varieties"

Olsson has some other papers that might be useful. He takes them off his web page when they get published, but sometimes you can find them with Google Scholar.

Edit: I haven't seen any work on the log-canonical rings of modular curves, but I don't really work in that area. You should allow poles of order n/2 for weight n forms, so for level 1, you get extra stuff like E₁₄/Delta.