First, correspondence theories of truth are generally associated with realism, not idealism. The point of a correspondence theory is that there is a correspondence between mental or linguistic representations and reality. Linguistic content is not necessarily mental (not for externalists) and in any case, having a correspondence implies that there is something that is not mental/linguistic to which mental/linguistic entities correspond. Realism is the position that assumes just that, and idealism denies it. Idealism is generally associated with coherence conceptions (if everything is mental then no correspondence is needed, only coherence between different ideas in a conceptual scheme).
It seems to me that correspondence theories are dominant today.
Coherence theories of truth seem to imply that truth is relative to a coherent conceptual scheme, and that in a sense, contradictory schemes are all equally true by their own standards. The constraints on truth are not strong enough: truth is not objective.
An alternative family of theories are pragmatic theories which claim that truth has to do with ideal utility, assertability, verifiability, ... These theories tie truth to our epistemic situation.
A first difficulty of these conceptions is to define an epistemic agent in a non anthropocentric way: who do we include in our epistemic community?
A second difficulty has to do with "ideal". Bare utility (or assertability) will not do because there are beliefs that are temporarily useful (or justified, or...) but not true. True beliefs should be useful in the long run, ideally that is, but the criteria of "ideal" utility are not easy to define. what do we mean by that?
A third difficulty is that these conceptions do not respect bivalence (a proposition is either true or false) since it is easy to formulate a proposition such that neither it or its negation is ideally useful (or justified...), for example the proposition that there exists a huge golden mountain that disappears whenever someone looks at it (or attempts to observe its effects). The proposition is not obviously meaningless: we understand what it says. A fundamental indeterminacy ensues.
However bivalence is a fundamental aspect of classical logic.
Correspondence truth eschews these difficulties. It seems to capture more adequately the intuition that truth transcends our epistemic position and capacities, and concerns something independent from us.
