How to decide which regularization (L1 or L2) to use?
Is there collinearity among some features? L2 regularization can improve prediction quality in this case, as implied by its alternative name, "ridge regression." However, it is true in general that either form of regularization will improve out-of-sample prediction, whether or not there is multicollinearity and whether or not there are irrelevant features, simply because of the shrinkage properties of the regularized estimators. L1 regularization can‘t help with multicollinearity; it will just pick the feature with the largest correlation to the outcome. Ridge regression can obtain coefficient estimates even when you have more features than examples... but the probability that any will be estimated precisely at 0 is 0.
What are the pros & cons of each of L1 / L2 regularization?
L1 regularization can‘t help with multicollinearity. L2 regularization can‘t help with feature selection. Elastic net regression can solve both problems. L1 and L2 regularization are taught for pedagogical reasons, but I‘m not aware of any situation where you want to use regularized regressions but not try an elastic net as a more general solution, since it includes both as special cases.