Is there any evidence for consumer utility-maximising behaviour, at individual or market level? Even though utility maximisation is ubiquitous in economic textbooks to model consumer behaviour, its usefulness is rarely demonstrated by evidence.


Is there any evidence that some consumers do behave that way on at least some markets?


Utility maximisation applied to a representative agent is also often used to model the demand side of a whole market. However, here also, this is hardly ever justified by evidence. On the other hand, alternative models seem to have good properties too: for instance in Complex Economics by Alan Kirman, it is said that random consumption + heterogeneity of income suffices to construct an aggregate demand that decreases as price increases. Random consumption does not seem more stupid than perfectly informed utility-maximisation... but sounds definitely simpler.


Is there evidence that utility maximisation used as a model of the demand side of a market is actually better than other (simpler) models of consumer behaviour?
Since utility by definition measures something very innate or subjective, it is difficult to obtain any non-experimental data on it. Hence researchers focus on choice, which is something observed in practice. This is where WARP, GARP and SARP comes in. Of which, GARP is the most relevant.
Any data generated by a utility maximising agent (with well behaved preferences) will satisfy GARP. The converse is true as well through Afriat's theorem - for any data satisfying GARP, there exists a utility function that rationalises it.
Note that it is difficult to study aggregate demand for such conclusions as well. The Sonnenchein-Mantel-Debreu Theorem basically kills any predictive power in this regard. So I would suggest you look up papers that empirically test GARP from choice data. Here's one such paper: Echenique et al
P.S. Mentioning @MichaelGreinecker here since I just noticed his comment that gave the same answer. :)