Usually you throw everything and see what sticks. There are some obvious trends like if you're doing image classification, a convnet will beat just about anything else. There are problems which are particularly well suited to e.g. random forests and cases where you might want to use a simpler model which will run faster at the expense of a bit of accuracy.

In terms of neural nets, people have more or less done the hard work for you. Typically you'll want to take an empirically well-performing network like VGG, ResNet, Inception, etc and then re-train the top few layers. There is ongoing work in the field to try and train the structure of the network as well.

If you look at Kaggle, the vast majority of winners do so using ensembles. In a recent example - Iceberg or Ship - the winning team used a bit of feature engineering and apparently over 100 ensembled CNNs:

> We started with a CNN pipeline of 100+ diverse models which included architectures such as customized CNNs, VGG, mini-GoogLeNet, several different Densenet variations, and others. Some of these included inc_angle after the fully connected layers and others didn’t.

https://www.kaggle.com/c/statoil-iceberg-classifier-challeng...

It's an ugly kitchen sink approach, but it works.