We present algorithms for supervised classification tasks in machine learning that are inspired by quantum mechanics. They rely on the so-called problem of quantum state discrimination, which consists in identifying which one of a known set of quantum states has been prepared based on the outcome of a quantum measurement on the state. After building the quantum states associated with the training data, which are encoded as density matrices, we explore various quantum measurement strategies suited for the classification task, including those based on semi-definite programming and on the so-called Pretty Good Measurement. We also consider how the measurement performs on multiple copies of the quantum state, since this in principle improves quantum state discrimination at the price of a higher computational cost. For the Pretty Good Measurement classifier, an analogue of the well-known Kernel Trick in learning theory can be devised, so that the cost scales only with the number of examples in the training dataset. This allows to study the performance of this classifier on tensor products of the quantum state. Finally, the different classifiers are benchmarked on the MNIST and MNIST-1D datasets.