I would say I believe that random mutations happen to some species maybe (in fact I have no way of knowing if any of them are truly random, but perhaps a few are).
Why wouldn't they be random?
Chemistry is random. It happens according to physical law, sure, but at a molecular level a reaction happens because of a completely chance interaction between two molecules in solution. We know that it's random because that's how molecules move in solution - randomly. We know
that because it's called "Brownian motion" and it's observed, directly, to be random.
Because these interactions are random, that can add randomness to a chemical reaction. For instance, the reaction of an asymmetric alkene with acid produces several different alcohols:
because the hydronium ion (H3O+) can add to the double bond (the two lines) at either end, and the product can re-arrange itself into a more stable form (the major product.) You'll always get these products, and always in a certain proportion when the reaction is performed at a specific temperature. In that sense, chemistry isn't random at all.
But if we could somehow zoom in on the reaction flask and see molecules interacting and reacting with each other, we wouldn't be able to
predict how any specific molecule was going to end up, because whether the hydronium attacks the double bond at the terminal end or the internal end is entirely random - it's simply a matter of how much specific kinetic energy those molecules collided with, and that energy is basically random.
All of chemistry is like this. Chemical reactions are reliable because they're
stoichiometric , that is, they follow statistical determinations. We can't say how any
particular 3-methyl-1-butene is going to react, but we can calculate with great precision how
many of them are going to form 2-methyl-2-butanol, 3-methyl-1-butanol, and 3-methyl-2-butanol.
All of this is the long way around to tell you that mutations of DNA are
also chemical reactions, they occur when mutagens interact with the DNA molecule, and
where they do that is a completely random thing. What they may do there is also random. Trying to mutate
a specific location in DNA in a specific way is such a difficult problem that it's taken us 40-50 years to learn to do it. Mutating DNA
randomly is incredibly easy, because randomness is the default state of chemistry. Getting chemistry to
not be random is very difficult indeed.
So that's how we know - we
know - that mutations are random - because mutation of DNA is a chemical reaction, and chemical reactions are random. Expose DNA to a mutagenic substance and it's inevitable that DNA will mutate.
Where and
how it will mutate is random, according to inviolable physical laws which you can test with a children's chemistry set.