Efforts to obtain explicit results from “effectively computable” results

This is mostly a number-theoretic question, but not necessarily so. In number theory there is a well-known dichotomy between “effective” and “ineffective” results. Many of the most famous ineffective results involve Siegel, though he did not originate many of them. For example his theorem on finiteness of integral points of curves of positive genus is famously ineffective, as well as his lower bound for the class number of imaginary quadratic fields. The ineffectiveness of the Thue-Siegel-Roth theorem is another famous example.

Ineffective means that, even in principle, the proof does not allow one to extract a finite (terminating) procedure to actually compute the finite set of solutions that the proof says must exist. On the flip side, “effective” means that a finitely-terminating procedure can be extracted or explicitly given by the proof, even if it is highly impractical. Many of the results obtained by Baker’s method for example are effective but not practical.

An even finer layer are explicit results, where all implied constants and exponents are given explicitly. There’s relatively little emphasis on this aspect in number theory, with occasional exceptions, most notably Helfgott’s proof of ternary Goldbach in its entirety, which involved a highly non-trivial sequence of explicit estimates.

Outside of number theory, there should be many examples of this type as well. One thing I can think of is computing a set of generators for the ring of polynomial invariants of some representation, i.e., GIT. To my understanding this is rarely done explicitly.

My question is: are there any active research programs that aim to turn “effectively computable” results into explicit results?