Ideal Use Cases for Splines In general, I have often heard of "splines" being referred to as "old models", criticized for being prone to overfit the data, and being considered to be only better than "higher order polynomials".

However, splines still seem to be used despite the shortcomings mentioned above.
My Question: Are there any ideal use cases for splines? Are there any industries/types of problems where spline based models are seen as the "dominant methodology" (e.g. survival analysis models for time-to-event data)?
I have heard that at times, spline models are able to "interpolate" data better than other types of statistical models - I have also seen a few authors attempt to use splines to model the "rate parameter function" in a poisson process.
Can anyone please provide comments on this?

Splines can be seen as non-parametric interpolation or fitting tools. So, the ideal application would be a case where you don't have a model to describe the variable but need to either interpolate it or produce a smooth version of the data.
Splines are often used in conjunction with other methods. For instance, take a look at linear splines: they model a function as a piece-wise linear function. If you think that your response has a nonlinear relationship to a variable, you could model the variable as a linear spline and enter into your OLS equation. See, mkspline function in State, for instance. You can think of GAM (generalized additive model) as an application of a similar approach, where you can apply a (non-smoothing) spline to a variable before throwing it into a regression.
Cubic (non-smoothing) splines are used to upsample variables, e.g. convert quarterly GDP variable into monthly for a model where the response is monthly.
You seem to describe the smoothing splines in your question, that's a different approach to interpolation altogether. It models the response variable as a low order polynomial piece-wise plus introduces the roughness penalty to make the output smooth.
Summarizing, there are many different spline applications used in the industry, some of them are stand-alone (like interpolation) and the others are integrated in other techniques such as regressions.
