Breathing is one of the body's most basic functions and abnormal breathing can indicate underlying cardiopulmonary problems. Monitoring respiratory abnormalities can help with early detection and reduce the risk of cardiopulmonary diseases. In this study, a 77 GHz frequency-modulated continuous wave (FMCW) millimetre-wave (mmWave) radar was used to detect different types of respiratory signals from the human body in a non-contact manner for respiratory monitoring (RM). To solve the problem of noise interference in the daily environment on the recognition of different breathing patterns, the system utilised breathing signals captured by the millimetre-wave radar. Firstly, we filtered out most of the static noise using a signal superposition method and designed an elliptical filter to obtain a more accurate image of the breathing waveforms between 0.1 Hz and 0.5 Hz. Secondly, combined with the histogram of oriented gradient (HOG) feature extraction algorithm, K-nearest neighbours (KNN), convolutional neural network (CNN), and HOG support vector machine (G-SVM) were used to classify four breathing modes, namely, normal breathing, slow and deep breathing, quick breathing, and meningitic breathing. The overall accuracy reached up to 94.75%. Therefore, this study effectively supports daily medical monitoring.