SVM vs logistic regression vs neural network

When to use which one?

First of all, logistic regression is very similar to svm with linear kernel and can be used interchangeably in practice .

Look at number of features (n) vs. Number of training example (m)
n >= m, u don't have many training example,  so no point in using complex algorithm to overfit the the small amount of data. Use logit /linear svm
n is small (say under 1k), m is intermediate (~10k or even more if you have enough resources / don't mind waiting. ..)
Gaussian svm so we can fit complex boundary
m is huge. Add features on your own and then use logit / linear svm

How about neural network?
Works for all the scenario,  just quite a bit slower