An exhaust system includes at least one exhaust gas purification component, at least one exhaust line, at least one connecting device for connecting the exhaust system to an internal combustion engine of a motor vehicle and at least one fastening element for additionally fastening the exhaust system to the motor vehicle. The purification component can oscillate at a first resonance frequency range of less than 150 Hz in a fastened state. The purification component can oscillate due to vibrations of the engine at least when the engine is started or switched off. This, in particular, allows ash residues to be removed from a purification component during starting of the motor vehicle, so that maintenance expense for the purification component is reduced and effective operation thereof is possible over the long term. A motor vehicle having an exhaust system and a method for operating the motor vehicle are also provided.