Autonomous structural health monitoring (SHM) of a large number of bridges became a topic of paramount importance for maintenance purposes and safety reasons. This article proposes a set of machine learning (ML) tools to perform automatic detection of anomalies in a bridge structure from vibrational data. As a case study, we considered the Z-24 bridge for which an extensive database of accelerometric data is available. The proposed framework starts from the stabilization diagram obtained through operational modal analysis (OMA) to perform the clustering of modal frequencies and their tracking by density-based time-domain filtering. The features extracted are then fed to a one-class classification (OCC) algorithm to perform anomaly detection. In particular, we propose two new anomaly detectors, namely, one-class classifier neural network (OCCNN) and OCCNN2, that find the normal class (the boundary of the features space in normal operating conditions) through a two-step approach: coarse and fine boundary estimate. The detection algorithms are then compared with known methods based on the principal component analysis (PCA), the kernel PCA (KPCA), the Gaussian mixture model (GMM), and the autoassociative neural network (ANN). The proposed OCCNN solution presents increased accuracy and F1 score over conventional algorithms, without the need to set critical parameters, while OCCNN2 provides the best performance in terms of F1 score, accuracy, and responsiveness.