MOSFET produces excessive spikes from transformer
First Note: You're talking about pure DC but I cannot see any high-value electrolytic capacitor (except 100n) for filtering after the bridge. I assume that you've forgotten to add that to the schematic in the question.

It's pretty normal to see spikes because there's switching. But please note that measurement technique is also important. Because when you use the ground clip of the probe, it acts as an antenna. This leads to some unwanted measurement artifacts. So, please make your measurement like shown below:



Note that even if the circuit does not contain any inductances, there can be seen some spikes/ringings caused by:


long cables
PCB design (i.e. long and thin tracks)
internal inductances of the load
internal drain inductance of the MOSFET


So, check these first. 

As for suppressing, you can put a pi filter between supply and the load. And also you can place a RC snubber across D-S terminals to suppress the spikes right at the switching node.