Why L2 loss is more commonly used in Neural Networks than other loss functions?

I'll cover both L2 regularized loss, as well as Mean-Squared Error (MSE):

MSE:

1. L2 loss is continuously-differentiable across any domain, unlike L1 loss. This makes training more stable and allows for gradient-based optimization, as opposed to combinatorial optimization.
2. Using L2 loss (without any regularization) corresponds to the Ordinary Least Squares Estimator, which, if you're able to invoke Gauss-Markov assumptions, can lead to some beneficial theoretical guarantees about your estimator/model (e.g. that it is the "Best Linear Unbiased Estimator"). Source: https://en.wikipedia.org/wiki/Gauss%E2%80%93Markov_theorem.

L2 Regularization:

1. Using L2 regularization is equivalent to invoking a Gaussian prior (see https://stats.stackexchange.com/questions/163388/why-is-the-l2-regularization-equivalent-to-gaussian-prior) on your model/estimator. If modeling your problem as a Maximum A Posteriori Inference (MAP) problem, if your likelihood model (p(y|x)) is Gaussian, then your posterior distribution over parameters (p(x|y)) will also be Gaussian. From Wikipedia: "If the likelihood function is Gaussian, choosing a Gaussian prior over the mean will ensure that the posterior distribution is also Gaussian" (source: https://en.wikipedia.org/wiki/Conjugate_prior).

2. As in the case above, L2 loss is continuously-differentiable across any domain, unlike L1 loss.