Is it methodologically valid to try multiple feature selection methods and report the best? I am working on a classification problem. I have used multiple feature selection methods, and a model is selected (i.e. hyperparameter optimization) and trained for each of the selected features. I then tested each of these trained models on a test dataset. 
I am now planning to publish the results, but the question is: can I only report the results achieved with the best feature selection method? I intent to state the fact that other feature selection methods have been used, but the reported one is the one with the best test performance. 
Does this undermine the validity of the results to be reported?
 A: You should absolutely report your full procedure.  Optimizing one hyper parameter using a hold out dataset is different from optimizing many, and then selecting the best (which can be viewed as a further hyper parameter optimization, over an indexing parameter of the feature selection methods).
Whether it undermines the validity of the results reported is extremely sensitive to exactly what you are reporting.  There's nothing wrong with optimizing many hyper parameters, but you must be honest about the exact procedure you followed, and what tradeoffs were incurred as a consequence.  The fundamental risk you expose your self to is, of course, overfitting, not just to your training data set, but also to your testing.
Certainly, if you make no mention of the discarded feature selection methodologies, you are doing your readers a disservice and misrepresenting your methodology.  I'm glad you were not considering this, as it is most definitely cheating.
If you report your final test set error of your best classifier as the expected out of sample error, that is also invalid, as you have used that test set to tune multiple hyper parameters.  This would force the test error estimate to have a considerable optimism bias.  This is certainly worth mentioning. It's always best to have a data set held out that you don't peek at until all is said and done.  If you have it, you should find your error rate on this data set, and report it.
If you are reporting frequentist summary statistics, p-values, confidence intervals of your best classifier, these are also biased optimistically.  The theory underlying these statistics assumes that you are fitting the one correct model, and not grid searching a parameter space (multiple times).
So my advice would be to report your methodology fully, understand the tradeoffs and risks, and inform your readership fully.
