A typical cartoon history of philosophy of science goes like this: (1) Carnap and the positivists had a theory of how hypotheses are verified; (2) Popper showed that they can only be falsified; (3) then Quine and Kuhn came along and showed that, actually, there can be no crucial experiments falsifying a theory because theories are complex combinations of propositions – so we have a messier process in which a paradigm can eventually be overturned if it proves unsuccessful in coping with a growing number of facts; (4) as a side-note to Kuhn, Feyerabend showed that there’s no such thing as a single scientific method, but you can usually ignore him as he was an anti-science crazy person.
This supposed intellectual progression of philosophers’ understanding of science is in many ways incorrect, partly because Carnap was in many ways better than Popper. A Bayesian version of the cartoon history goes like this. First, you can safely ignore Popper because he didn’t understand probability theory, and, overall he's largely a distraction. There’s nothing wrong with saying that a theory is “verified” if by that you simply mean that it is the theory most consistent with the available information. (Popper built a whole theory of “propensities” in the attempt to argue against Bayesian subjective probabilities. By contrast, Carnap, who was Bayesian, tried to convince his fellow positivists to abandon Richard Mises’s radical nonsense about objective probabilities. He failed in being persuasive, but we can say that he has now been vindicated by Edwin Jaynes and the Bayesian revolution.) Furthermore, Quine and Kuhn can be seen as expanding on Carnap’s critique of his fellow positivists from “Empiricism, Semantics, and Ontology”. The positivists were at the time arguing that one should not use concepts that are not empirically measurable (e.g. Jordan wrote an entire quantum mechanics textbook purely in terms of measurables and purging all talk of “wave functions” etc.). Carnap argued that this was a ridiculously strong requirement, and that it was perfectly ok for science to use “conceptual frameworks” which do not involve direct empirical measurements – if the framework is not useful, it will eventually be replaced:
To decree dogmatic prohibitions of certain linguistic forms [conceptual frameworks] instead of testing them by their success or failure in practical use, is worse than futile; it is positively harmful because it may obstruct scientific progress. The history of science shows examples of such prohibitions based on prejudices deriving from religious, mythological, metaphysical, or other irrational sources, which slowed up the developments for shorter or longer periods of time. Let us learn from the lessons of history. Let us grant to those who work in any special field of investigation the freedom to use any form of expression which seems useful to them; the work in the field will sooner or later lead to the elimination of those forms which have no useful function. Let us be cautious in making assertions and critical in examining them, but tolerant in permitting linguistic forms. [emphasis in the original]
If this sounds very Kuhnian to you, it's because it is. If this reminds you of Elinor Ostrom’s distinction between frameworks, theories, and models, congratulations! Quine further strengthened Carnap’s critique by showing that there are no such things as purely formal propositions – anything that’s meaningful has some (however strenuous) connection with facts. In other words, there was no point in Carnap’s futile attempt to keep the distinction between purely formal and purely empirical (and, worst, try to build a theory of meaning on it). This is important because it also means that all "empirical facts" are unavoidably interpreted through some theoretical lens. There's no way to escape theory. We always have only combinations of theory and facts. Kuhn can be seen as simply providing a stylized historical account of how Carnap’s frameworks (what he called “paradigms”) did indeed change from time to time. He’s important because he brought history and facts to a philosophical fight, which, of course, no positivist could object to.
Now Feyerabend becomes a much more important figure because he showed that scientists have never been and are still nowhere close to the idealized rational Bayesians that the Carnap-Kuhn picture of science assumes. Kuhn’s history is still overly-beautified and whig. The latest version of the Bayesian picture of science can be found in Jaynes’s book Probability Theory: The Logic of Science, and a summary of it is in his paper on “Search Theory” (see esp. the conclusion). The problem with this picture is that it assumes that paradigms/frameworks change on a purely rational basis. Facts accumulate, and failing frameworks get changed. Feyerabend showed this wasn’t exactly so. You can see him as a sort of behavioral economist analyzing how scientists behave. This is why he is often misinterpreted as being “anti-science”. But far from being anti-science he is actually explicit that he takes it as given that scientific progress has in fact occurred, and he wants to explain how that was possible. His whole thing is that he’s concerned that philosophers of science will inadvertently ruin it (echoing Carnap’s concern and tolerance cited above). He says that his account works regardless of how you define “progress”.
Feyerabend basic puzzle is this: (1) Let’s agree that “scientific progress” has occurred, even if we disagree about the exact details about how to define this progress. (2) Here’s evidence that scientists are not rational Bayesians and, hence, scientific progress didn’t happen simply as an inherently rational process (as assumed by Carnap or Kuhn or Jaynes). (3) If (1) and (2) are correct, how does scientific progress actually happen? What's the mechanism that actually drives this progress if (2) is at best only part of the answer. His answer is that we must search for an institutional explanation. This means that Michael Polanyi and Tullock have more important things to say about why science works than the regular philosophers of science.
I think we are now much further along the way to understand the institutions that make science work. There's now a fairly large literature on the economics of science looking at scientific entrepreneurship and how prestige works. Here’s my paper, building on Smolin, arguing that science can be expected to work (despite individual scientists' failures) as long as its polycentric organization is preserved. Here’s Boettke, Coyne and Leeson’s paper on how important things can get lost when switching from one paradigm to the next.