How do chemists verify certain phenomena like 1,2-hydride shifts, 1,2-methyl shifts and ring expansions? I've been learning about organic chemistry independently on a ship without an instructor or university resources like a lab setting for the past year. I was wondering how do chemists verify that their mechanisms are correct?

For example, take 3-methylbutene and add the reagent HCl. From the 1,2-hydride shift we get 3-chloro-3-methylbutane. For a 1,2-methyl shift take 3,3-dimethylbutene with the same reagent HCl. We know that from the mechanism we get 2-chloro-2,3-dimethylbutane. But how do you verify that these are the major products?
I wanted to put this as a comment but I don't have enough rep :p

It happens the other way round, first the products are obtained, then the mechanism is proposed.

The point is, according to me that once it has been verified which product is major and which minor, chemists may try to come up with a suitable explanation for the product obtained, on paper. They may try some things, come up with possible mechanisms and sometimes even phenomena like hydride shift, methyl shift etc. Once they come up with such an explanation, they try to apply it to other related problems and upon reaching the right solution every time, they can say this is what might have happened to a high degree of certainty. 

So you see, chemistry is also sometimes glorified problem solving :p