Is feature selection agnostic to regression/classification model chosen?
It depends on the algorithm you use to select features.  Filter methods that are done prior to modeling are of course agnostic, as they're using statistical methods like chi-squared or correlation coefficient to get rid of unnecessary features.
If you use embedded methods where features are selected during model creation, it is possible that different models will find value in different feature sets.  Lasso, Elastic Net, Ridge Regression are a few examples of these. 
It's worth noting that some model types perform well with sparse data or missing values while others do not.
So as the title suggests, my question is whether feature selection algorithms are independent of the regression/classification model chosen. Maybe some feature selection algorithms are independent and some are not? If so can you name a few of each kind? Thanks.