Introductory material on splines I am looking for a basic, step-by-step introduction into modelling with splines.
(I have encountered splines while teaching another topic. The textbook I am using does not cover splines in sufficient detail for the students to be able to replicate things by themselves. I hope there is some good introductory material out there so that I do not have to reinvent the wheel and write my own notes on the topic.)

This answer is coming from a biostatistics angle.
I would second Frank Harrell's Regression Modelling Strategies, as per Stephan's answer https://link.springer.com/book/10.1007/978-3-319-19425-7.
Other resources I have used include a 2010 paper on general implementation (1). Per the title, it is focused on restricted cubic splines and includes a SAS macro (I think SAS has a bit more built-in functionality nowadays, but I use R).
I also read a more recent paper (2) focused on discussing different spline types and then considering their implementations in R. This is quite handy since there is some review therein about maturity of different packages for fitting splines. For me this seems reasonably accessible at a conceptual level (for those wanting to consider and use splines in applied settings) while also containing sufficient detail on basis forms for those who prefer a mathematical presentation.
(1) Desquilbet L, Mariotti F. Dose-response analyses using restricted cubic spline functions in public health research. Stat Med 2010;29(9):1037-57. doi: 10.1002/sim.3841 https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.3841
(2) Perperoglou, A., Sauerbrei, W., Abrahamowicz, M. et al. A review of spline function procedures in R. BMC Med Res Methodol 19, 46 (2019). https://doi.org/10.1186/s12874-019-0666-3
https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-019-0666-3
