The use of fibre-reinforced plastics (FRPs) in various industrial applications continues to increase thanks to their good strength-to-weight ratio and impact resistance, as well as the high strength that provides engineers with advanced options for the design of modern structures subjected to a variety of out-of-plane impacts. An assessment of the damage morphology under such conditions using non-destructive techniques could provide useful data for material design and optimisation. This study investigated the damage mechanism and energy-absorption characteristics of E-glass laminates and sandwich structures with GFRP face sheets with PVC cores under quasi-static indentation with conical, square, and hemispherical indenters. An acoustic emission (AE) technique, coupled with a k-means++ pattern-recognition algorithm, was employed to identify the dominant microscopic and macroscopic damage mechanisms. Additionally, a post-mortem damage assessment was performed with X-ray micro computed tomography and scanning electron microscopy to validate the identified clusters. It was found that the specific energy absorption after impact with the square and hemispherical indenters of the GFRP sandwich and the plain laminate differed significantly, by 19.29% and 43.33%, respectively, while a minimal difference of 3.5% was recorded for the conical indenter. Additionally, the results obtained with the clustering technique applied to the acoustic emission signals detected the main damaged modes, such as matrix cracking, fibre/matrix debonding, delamination, the debonding of face sheets/core, and core failure. The results therefore could provide a methodology for the optimisation and prediction of damage for the health monitoring of composites.