I agree with you however I may be even more cynical.  I feel people (almost everyone) checks for the normality of their dependent variable.  However I am under the assumption that many do not check for the normality of their predictor variables.  Furthermore, I would not be surprised if a very small minority did check for equal population variance.  

Level of measure are also very important assumptions or rules that must be followed.  Perhaps not so much in peer reviewed journals but at the conference level and in unpublished papers people quite  often ignore the fact that almost all statistical tests (beyond simple descriptions of the data) require continues variables.  I quite often see ANOVAs, and t-tests conducted with nominal/categorical variables.  

Is there a problem here?  Yes and no.  In my field, Criminology, the answer is mostly no as most published studies use data from large nationally representative samples that come from organizations including the federal government that have reputations of gathering and producing high quality representative data.  

I do feel that there are other fields that get data from smaller portions of the populations and thus they cannot be certain if their data is representative. However it wold be irresponsible for me to speak about a filed that I am not part of.  

If one is fitting a statistical model (using ANOVA or ReML) with Y as response and X's as explanatory / predictor variables, the statistical assumptions of normality and equal variances apply to residuals after fitting the model - not to the Y or the the X variables.

I always check the residuals after a model is fitted by some diagnostic plots.

I always check assumptions or some important issues (e.g., residuals, normality,hazard functions that are proportional over time)  to assure that I have a rigorous model and to decide/find an appropriate statistical method.

I agree with you however I may be even more cynical.  I feel people (almost everyone) checks for the normality of their dependent variable.  However I am under the assumption that many do not check for the normality of their predictor variables.  Furthermore, I would not be surprised if a very small minority did check for equal population variance.  

Level of measure are also very important assumptions or rules that must be followed.  Perhaps not so much in peer reviewed journals but at the conference level and in unpublished papers people quite  often ignore the fact that almost all statistical tests (beyond simple descriptions of the data) require continues variables.  I quite often see ANOVAs, and t-tests conducted with nominal/categorical variables.  

Is there a problem here?  Yes and no.  In my field, Criminology, the answer is mostly no as most published studies use data from large nationally representative samples that come from organizations including the federal government that have reputations of gathering and producing high quality representative data.  

I do feel that there are other fields that get data from smaller portions of the populations and thus they cannot be certain if their data is representative. However it wold be irresponsible for me to speak about a filed that I am not part of.  

I saw too much people did not check that and even never hear and knew what is that. Most journals do not ask them to provide the evidences. Hence, we can not easy tell their statistical models are suitable or not. This is a very terrible thing in the field of research, because people only use the easy software package but not understand what they really did.

I favor methods with few assumptions. I tend not to check the assumptions.

So I check for normality. My choices are 1) reject the null hypothesis and try a different statistical technique, or 2) fail to reject the null hypothesis, recognize that I could reject the null hypothesis with a larger sample size, and therefore behave as if the null hypothesis was true. This game seems dumb.

So I check for equal variances. I have tried randomization methods, and I typically am able to reject the null hypothesis. If I fail to reject the null hypothesis, I figure that I could solve the problem by increasing my sample size. This game seems suspiciously similar to the normality testing game.

Having successfully rejected all of the null-hypotheses for all the tests of model assumptions, I can no longer use any current statistical method. I have a pile of numbers. I don't much like that outcome.

A more fun game is to run the analysis using several techniques with somewhat different assumptions. Unfortunately, one can spend a great deal of time with this game as one tries to figure out why the three different methods gave three different (hopefully not mutually exclusive) answers. Alternatively, I can restrict my data collection activities to answering the simplest of questions where no one will raise awkward questions. Life is explained by multiple comparison procedures: more bars in more places. I am not particularly good at developing super-simple experiments.

I always spend some time looking at the residuals in a regression analysis.

So now one asks, how many research reports in the library used inappropriate statistical methods or had too little data to properly test the assumptions? Something like 100%. Yet for those problems it would seem that we have made some progress relative to the stone knives and bear skins of our distant forefathers. While any given report may be flawed, it is repeated in different ways. There is progress.

The other problem is that some of the statistical methods are "robust to moderate departures from normality." What does that mean? I need to check for symmetry, and Skewness=0, Kurtosis=0, but if skewness is between -0.5 and +0.5 everything will be ok? Robust is never quantified, but then that would probably involve another null hypothesis test.

I confess that I partly hate this answer. It is too easy for someone to take a "do whatever you want" approach. So try this: What do you want from Science? Is this a game to con other people so that you can get as much money as possible? Is this an attempt to better understand the world around you and assist others in a better understanding of their world? If the latter, then one tries to satisfy as many assumptions as possible, use methods that overcome/avoid the problems, and try different approaches that may produce results that you don't like. To the best of your ability perform experiments and analyses in such a way that it is likely that others will get a similar outcome. So it is critical that everyone understand that there are assumptions. It is also important to know that some methods are inappropriate because the assumptions are not satisfied. That leaves a number of possibilities that are more or less correct depending on exactly how you view the problem, and what questions are being asked.

It is worth adding that some assumptions cannot be tested, though they can be defended (based on whatever knowledge you have handy). Pearl talks about this a lot with respect to causal diagrams and it is also relevant to defending priors. Other examples exist too (e.g., deciding between Missing at Random and Missing Completely at Random).

There is also the problem of using data from a small study to "test" assumptions for that study. You are still needing to assume a lot even if your sample skewness (or whatever) statistics are close to zero.

Thanks for these very useful suggestions and comments on your own practices for what I asked about.