The growing number of man-made debris in Earth's orbit poses a threat to active satellite missions due to the risk of collision. Characterizing unknown debris is, therefore, of high interest. Light Curves (LCs) are temporal variations of object brightness and have been shown to contain information such as shape, attitude, and rotational state. Since 2015, the Satellite Laser Ranging (SLR) group of Space Research Institute (IWF) Graz has been building a space debris LC catalogue. The LCs are captured on a Single Photon basis, which sets them apart from CCD-based measurements. In recent years, Machine Learning (ML) models have emerged as a viable technique for analyzing LCs. This work aims to classify Single Photon Space Debris using the ML framework. We have explored LC classification using k-Nearest Neighbour (k-NN), Random Forest (RDF), XGBoost (XGB), and Convolutional Neural Network (CNN) classifiers in order to assess the difference in performance between traditional and deep models. Instead of performing classification on the direct LCs data, we extracted features from the data first using an automated pipeline. We apply our models on three tasks, which are classifying individual objects, objects grouped into families according to origin (e.g., GLONASS satellites), and grouping into general types (e.g., rocket bodies). We successfully classified Space Debris LCs captured on Single Photon basis, obtaining accuracies as high as 90.7%. Further, our experiments show that the classifiers provide better classification accuracy with automated extracted features than other methods.