Scientific realism is the view that science seeks the truth and sometimes finds it. This is a pretty commonsensical view and is probably the instinctive philosophy of most working scientists. The "No Miracles Argument" (henceforth "NMA") was most famously presented by Hilary Putnam. It has since been embellished by other philosophers.
(P1) Science has progressed (or is successful). Science has advanced the sum of our knowledge of the things around us, and has advanced our ability to manipulate those things. Science is, by and large, successful in predicting how those things are going to behave in the future, letting us be so successful in manipulating them.
(P2) The (approximate) truth of our most successful scientific theories is the best explanation of the progress/success of science. Scientific realism provides us with a better explanation for this progress/success than any other philosophy of science.
(C) Therefore, all other things being equal, our most successful scientific theories are more probably (approximately) true than not.
By "most successful" I reference those scientific theories that have a proven track record of successfully predicting, with great accuracy, the future behavior of our real world furniture. Our highly technological life-style depends on that record.
Some people do debate premise (P1), especially the notion that the history of science demonstrates "progress". But any debate on this issue must deal with the apparent fact that we now are much more capable of manipulating our environment, and the things in it, to suit our purposes than we were even a few decades ago. By most accepted standards this is "progress" and "success". For this short essay, I will not comment further on this debate.
The NMA rests on the validity of an inductive inference to the best explanation (henceforth "IBE"). As such, it is dependent on our notion of what constitutes a "best" explanation. Since there is no commonly accepted criteria for which of competing explanations is the "loveliest" (in Peter Lipton's word), there is room to debate what is actually the "best" explanation for the progress/success of science. Some criteria of "loveliness" can be devised to show that, for example, that "God's Will" is a lovelier explanation. For this short essay, I will not comment further on this part of the debate.
One must keep in mind the context of the argument - a debate between scientific realists and scientific anti-realists. It therefore presupposes a somewhat realist metaphysics -- a metaphysics where all parties agree that the objects studied by science are independent of how or whether we think about them. (I say "somewhat realist" to include Berkelean Idealism, as an example.) When the argument is used in an effort to support metaphysical realism, it is properly guilty of begging the question. IBE presupposes a realist metaphyics. Two of the main targets of NMA are Instrumentalism (theories about unobservable phenomena are regarded as having no scientific meaning), and the anti-realism of Van Fraassen ("Anti-realism is a position according to which the aim of science can well be served without giving such a literally true story, and acceptance of a theory may properly involve something less (or other) than belief that it is true.") Both of these accept metaphysical realism.
Most of the criticism of the NMA is based on challenging the premise of IBE that a hypothesis that is judged "best" by some criteria is more likely to be true than not. This disrupts the link from the premises to the conclusion (C) above. The two main challenges here are the "Pessimistic Induction" argument, and Van Fraassen's Darwinian alternative.
The Pessimistic Induction argument is a reductio ad absurdum argument that demonstrates that the basic premise of NMA that success is a reliable test for truth is not supported by the history of science.
(1) Assume that the success of a theory is a reliable test for its truth.
(2) Most current scientific theories are successful.
(3) So most current scientific theories are true.
(4) Then most past scientific theories are false, since they differ from current theories in significant ways.
(5) Many of these false past theories were successful.
(6) So the success of a theory is not a reliable test for its truth.
But if we employ a pragmatist notion of "true-enough" instead of a more absolute notion of true imported from deductive logic, this pessimistic induction argument fails. An hypothesis or a scientific theory is "true-enough" if and only if it is empirically successful -- if it results in predictions and deductive entailments that correctly describe our experiences within the accuracy of our empirical requirements . Van Fraassen comes very close to this pragmatic notion of truth with his "empirical adequacy". But he does not recognize that the pragmatic notion he defines as the proper concept of "truth" for an inductive reasoning context. If "true-enough" is employed, then clause (4) above is historically false. Most past scientific theories were true-enough at the time, even though they are no longer true-enough because our requirements for empirical accuracy have changed.
Van Fraasen's "Darwinian" alternative argues that empirical adequacy or instrumental success is sufficient to account for the success of science. His reason is that scientific theories are constructed to be empirically adequate/instrumentally successful by having to deal with the anomalies of predecessor theories. Later theories must therefore be more adequate/successful. So a form of Darwinian selection guarantees that later theories are "fitter" than their predecessors in exactly these respects. There is no need to invoke the "truth" of these theories. But the Darwinian alternative ignores the question of why later theories are more adequate/successful than earlier ones. What makes later theories turn out to be more adequate/successful if it is not their asymptotic approach to "truth"?
Only if one is already a scientific realist to begin with will one accept that the Inference to the Best Explanation for the success of science is that scientific realism is true. If one is not already a scientific realist to begin with, then there are enough unresolved problems with both IBE and the pragmatist notion of "true-enough" involved, that the No-Miracles argument will not be obviously persuasive.