Does anyone know around what year MCMC became commonplace (i.e., a popular method for Bayesian inference)? A link to the number of published MCMC (journal) articles over time would be especially helpful.
What can be reasonably seen as the first MCMC algorithm is what we now call the Metropolis algorithm, published by Metropolis et al. (1953). It emanates from the same group of scientists who produced the Monte Carlo method, namely, the research scientists of Los Alamos, mostly physicists working on mathematical physics and the atomic bomb.
The Metropolis algorithm was later generalized by Hastings (1970) and his student Peskun (1973,1981)
Although somewhat removed from statistical inference in the classical sense and based on earlier techniques used in Statistical Physics, the landmark paper by Geman and Geman (1984) brought Gibbs sampling into the arena of statistical application. This paper is also responsible for the name Gibbs sampling
In particular, Geman and Geman (1984) influenced Gelfand and Smith (1990) to write a paper that is the genuine starting point for an intensive use of MCMC methods by the main-stream statistical community. It sparked new inter-est in Bayesian methods, statistical computing, algorithms and stochastic processes through the use of computing algorithms such as the Gibbs sampler and the Metropolis–Hastings algorithm.
Interestingly, the earlier paper by Tanner and Wong (1987) had essentially the same ingredients as Gelfand and Smith (1990), namely, the fact that simulating from the conditional distributions is sufficient to asymptotically simulate from the joint.This paper was considered important enough to be a discussion paper in the Journal of the American Statistical Association, but its impact was somehow limited, compared with Gelfand and Smith (1990).
I couldn't find the number of journal articles published over time, but here is a Google Ngram plot for the number of mentions over time. It more or less agrees with the notion that MCMC became commonplace after the 1990 paper of Gelfand and Smith.
The excellent answer by knrumsey gives some history on the progression of important academic work in MCMC. One other aspect worth examining is the development of software to facilitate MCMC by the ordinary user. Statistical methods are often used mostly by specialists until they are implemented in software that allows the ordinary user to implement them without programming. For example, the software BUGS had its first release in 1997. That does not appear to have changed the growth trajectory in the N-Grams plot, but it may have been an influence in bringing the method into common usage among those users who found it intimidating to program their own routines.