Molecular featurisation refers to the transformation of molecular data into numerical feature vectors. It is one of the key research areas in molecular machine learning and computational drug discovery. Recently, message-passing graph neural networks (GNNs) have emerged as a novel method to learn differentiable features directly from molecular graphs. While such techniques hold great promise, further investigations are needed to clarify if and when they indeed manage to definitively outcompete classical molecular featurisations such as extended-connectivity fingerprints (ECFPs) and physicochemical-descriptor vectors (PDVs). We systematically explore and further develop classical and graph-based molecular featurisation methods for two important tasks: molecular property prediction, in particular, quantitative structure-activity relationship (QSAR) prediction, and the largely unexplored challenge of activity-cliff (AC) prediction. We first give a technical description and critical analysis of PDVs, ECFPs and message-passing GNNs, with a focus on graph isomorphism networks (GINs). We then conduct a rigorous computational study to compare the performance of PDVs, ECFPs and GINs for QSAR and AC-prediction. Following this, we mathematically describe and computationally evaluate a novel twin neural network model for AC-prediction. We further introduce an operation called substructure pooling for the vectorisation of structural fingerprints as a natural counterpart to graph pooling in GNN architectures. We go on to propose Sort & Slice, a simple substructure-pooling technique for ECFPs that robustly outperforms hash-based folding at molecular property prediction. Finally, we outline two ideas for future research: (i) a graph-based self-supervised learning strategy to make classical molecular featurisations trainable, and (ii) trainable substructure-pooling via differentiable self-attention.