The most compelling case that I think I’ve seen so far for anything resembling mutations that are non-random with respect to fitness (biased towards increased fitness) are outlined in these 2 papers:
To roughly summarise their results: cells growing on glycerol are more likely to experience an insertion of IS5 specifically upstream of a glycerol metabolism (glpFK), which acts as a promoter to upregulate glycerol metabolism. The mechanism for this is that glycerol binds to a repressor of the operon (glpR). This is what activates the operon in the first place by stopping glpR from binding to a control region. It turns out that having glpR bound to that control region stops IS5 insertions, so removing it (by any means) increases the prevalence of IS5 insertions. Another factor is the presence of cAMP: less cAMP = less binding of a cAMP complex to that control region which exposes it to even more to IS5 insertion.
Basically the presence of glycerol and absence of cAMP causes a very high insertion rate of IS5 into the control region to activate the operon. This can be advantageous in conditions where glycerol and a cAMP depressing toxic substance (such as 2-deoxyglucose, synthesised naturally by many organisms as a defence mechanism) are present simultaneously.
It’s also possible that this insertion is reversible, so the whole thing could be a temporary solution to the problem. This in effect makes it just like a “normal” mechanism of genetic regulation, little different in concept to any other operon that’s activated in response to a particular molecule. In other words, it’s readily evolvable. It would be striking if this kind of temporary regulation really was caused by an insertion and subsequent deletion of a particular transposon in a particular place. Although I don’t think this example, if accurate, would do much to make us reconsider the fundamentals of evolution, it is interesting.