Question: Consider our world at 18-19 centuries. Our civilization knows how to print books, build steam machines, we have made first experiments on electricity. And one day the [Mad God Sheogorath](http://elderscrolls.wikia.com/wiki/Sheogorath) is emerged. He can bend the reality by his will. 

For example, he can violate the principles of conservation of energy - build Perpetuum Mobile just to irritate scientists. 

He can convert Darwin back into the monkey just to help him prove his theory. 

He can make forgery of dinosaur skeletons, so we have found caterpillar riding tankosaurus instead of diplodocus.

He can resurrect people.
And, do a lot of things we are not aware.

So, in potential he can spoil every scientific acquired experiments result in a way, that is hard to see. 

So, the question is: is scientific method still possible in this world?

UPD: after 20 years Sheogorath have just vanished. Can we undo his trickery and restore science methods?

Answer: 
*Is the scientific method still possible in this world?*  

More or less.

The scientific method depends heavily on being able to prove things are either true or false (at least it used to) with as many few "maybe" in between as possible. Those "maybe" being mostly regarded as proof of falsehood.

Simply put, it depends on what has the Mad God touched.  

If all it does is punctually changing one thing, then scientific method can still be applied to everything else, with the warning that this all powerful being can have messed with it for fun and giggles.  

So the scientific method would take exceptions into account and distrusts single fact as proof - one way of another - even more than today.

The problem would come if the mad god changed some fundamental things so that they permanently gave random results. If she made it so that light passing through a prism changed color randomly, all science related to this particular field would be devastated.

*After 20 years Sheogorath have just vanished. Can we undo his trickery and restore scientific methods?*  

It depend directly on the permanence of the effects : killing zombies, deciding that once occurring tankosaurus are proof of nothing is easy.  

Restoring laws of physics bent by the will of a god is another matter.

*A short conclusion :*

All in all it could be regarded as pranks from a couple ten year old mischievous, all powerful kids.  

Fear and annoyment would be the main feelings toward the results.

Oh, and hate of those responsible.
If it's linked to a scientific group, then the whole scientific community could be seen as responsible, thus creating a huge problem.
As a writer and reader, if I'm dealing with a being with what we might call "godlike powers", they still have to operate within a larger reality. For example in your setup, Sheogorath builds a tiny "Perpetual Motion Machine" but is actually cheating because in some mysterious god power way, it is perfectly converting a few molecules of MysteriousInvisibleSubstance to Energy at the nano-level. Maybe there's even enough MysteriousInvisibleSubstance to last a thousand years and it's really really hard to detect. MysteriousInvisibleSubstance can also alter visible reality so that Darwin is now stuck in what appears to be a monkey body. Sheogorath can also in some odd way travel in time and use his nifty MysteriousInvisibleSubstance powered MatterReasssembler to screw with the fossils just to screw with the human's of the future's heads. Then Sheogorath disappears. 

Until the sentients are clever enough to figure out how to detect and then use MysteriousInvisibleSubstance at the molecular level, no "laws of reality" in that fictional world disturbed. That said, for the WorldBuilding exercise to be any fun to read, etc. When the sentients or others of Sheogorath's species figure out what he's been up to and how, Sheogorath is going to be in BIG BIG trouble, right?