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The support of gfortran for various standards is described at Standards (The GNU Fortran Compiler) and Major changes in GCC Gfortran by version | Scientific Computing and for Fortran 2018 at Fortran2018Status - GCC Wiki. The latest gfortran is gfortran 14, which will implement more features than your current GCC-11.3.0. You are probably safe using features through Fortran 2008, but the best answer will be obtained from trial and error. You can install more than one version of gfortran on your PC, and if a later version supports a feature you want to use, you could ask the system administrators to install a later version of gcc on the supercomputer.
Using a standard later than Fortran 90 will not stop you from using GSL and the other packages you listed, and in fact the C interoperability features of Fortran 2003, implemented in gfortran, enable you to call GSL in a standard-conforming way: FGSL 1.6.0 released