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Thumping noises on a car when it starts up could be caused by any number of things. The safest option is to not second guess what the problem is but take your vehicle into a workshop to be looked at.
Bad shocks or struts is one possibility.
Most likely a bad tire.
You can use a car noise database to compare noises at mycarmakesnoise.com
You are first going to hear some thumping and other strange noises coming from under there when you hit the brakes and let them off while your driving. What it does will range depending on how bad they are and what kind of car but, you will also probably notice it pulling to one side while accelerating after decellerating. if they are bad on both sides your gonna have steering control on the vehicle.
Dogs make howling noises in cars.
Maybe that thumping noise is the sound of another car smashing into the rear of your car.
Have your alignment checked.
Is it accompanied by a shuddering or vibration? You could likely have bad brake rotors.
Possibly a broken shock or bad stabilizer bushings
The frist thing I would check would be the engine mounts.
No.