One of the basic figures you get when running multiple linear regression using almost any off-the-shelf software is the F statistics. However, I cannot recall one situation, where the F value was low enough that I could say, the ratio of model MSE and sample variance was not significant. I do understand that the figure makes sense if we compare two competing models, but the way it is usually reported by most software is by measuring the relative decrease in variance. Maybe I am overlooking something, but why report the F statistic at all?
The F-test may give you some useful information in some cases. For example, sometimes we find that according to the t-tests the regressors are individually not significant, while the F-test rejects the null that all the regressors (except the intercept) are jointly not significant. This may be a sign of multicollinearity among the regressors, which will lead to higher standard errors of parameter estimates and larger confidence intervals and should be somehow addressed (for example, creating a proxy variable that is a combination of some of the regressors).