My view is that at least in (applied) econometrics, it is more and more the norm to use the robust or empirical covariance matrix rather than the "anachronistic practice" of relying (asymptotically) on the correct specification of the covariance matrix. This is of course not without controversy: see some of the answers I linked here at CrossValidated, but it is certainly a clear trend.
Examples include heteroscedasticity-robust standard error (Eicker-Huber-White standard errors). Some researchers such as Angrist and Pischke apparently advise always using heteroscedasticity-robust standard error rather than the "anachronistic" procedure to use normal standard error as default and check whether the assumption $E[uu'] = \sigma^2 I_n$ is warranted.
Other examples include panel data, Imbens and Wooldridge write for example in their lecture slides argue against using the random effects variance covariance matrix (implicitly assuming some misspecification in the variance component as default):
Fully robust inference is available and should generally be used. (Note: The usual RE variance matrix, which depends only on $\sigma_c^2$and $\sigma_u^2$, need not be correctly specified! It still makes sense to use it in estimation but make inference robust.)
Using generalized linear models (for distributions which belong to the exponential family), often it is advised to use always the so-called sandwich estimator rather than relying on correct distributional assumptions (the anachronistic practice here): see for example this answer or Cameron referring to count data because pseudo-maximum likelihood estimation can be quite flexible in the case of misspecification (e.g. using Poisson if negative binomial would be correct).
Such [White] standard error corrections must be made for Poisson regression, as they can make a much bigger difference than similar heteroskedasticity corrections for OLS.
Greene writes in his textbook in Chapter 14 (available on his website) for example with a critical note and goes more into detail about the advantages and disadvantages of this practice:
There is a trend in the current literature to compute this [sandwich] estimator routinely, regardless of the likelihood function.* [...] *We do emphasize once again that the sandwich estimator, in and of itself, is not necessarily of any virtue if the likelihood function is misspecified and the other conditions for the M estimator are not met.