Just as you said 'different statisticians have different ideas'. But the regression model only can abstractly discover the relationship between independent variable and dependent variable, because there is no independent variable has no relationship with dependent variable. We need to find the most important independent varibale with dependent,so it is necessary to omit some insignificant independant variables ,and it is not wise to promote
R-squared ---- just as doc@duck said 'when your model has great than .9 with R-squared,
the data would be skeptical' . So it is enough to find several important independent variables with dependent variable.
About 'indications of curvature or non-linearity.' , I think that using 'plot student.*x' statement to find the residual whether ~N (0,sigma^2),
if it ~N (0,sigma^2,then Model fit these data very well, if not can add x^2 or x^3 to see whether enhance the fitness of model and data.
The above all is just my opinion.:-)
Ksharp