Options for GCC's Fortran compiler
By default, the gfortran compiler issues some warnings, but it can produce more. It is always good practice to switch on "all warnings" with -Wall. Contrary to expectations, -Wall doesn't activate all warnings though. A number of extra warnings on top of those enabled by -Wall can be switched on by specifying -Wextra. It is highly recommended to always use these two compiler options, and eliminate all warnings reported by them, i.e.,
$ gfortran -Wall -Wextra ...
Language specification conformance
The gfortran compiler can also check for language specification conformity, at least up to some level, to quote the documentation, "improvements to GNU Fortran in this area are welcome." The -Wpedantic flag will activate this, and you should specify the specification it should check, i.e.,
-std=f95-std=f2003-std=f2008-std=f2018
For example, to check for Fortran 2003 compliance, use
$ gfortran ... -Wpedantic -std=f2003 ...
Additional warnings
A few other options can be helpful as well. Those are not activated by -Wall and -Wextra.
It is good practice to have implicit none in each compilation unit, but that is also easy to forget. The compiler has a flag, -fimplicit-none, that will verify that all variables have been declared, regardless of whether implicit none was specified.
Another good practice is to explicitly declare what to use from a module in a use statement. The compiler can warn you when this has not been done if you specify the -Wuse-without-only flags.
You may want to be warned if you use variables that are not uninitialised, since this could lead to nasty bugs. Use the -Wmaybe-uninitialized flag to do so (included in -Wall). Note: this warning will only be reported when the compiler is optimising the code, i.e., when you specify at least -O1. Without optimisation, enabling this warning would produce too many false positives.
It can also be useful to verify that procedures are either intrinsic, or have been declared external explicitly. You can activate this check using the -Wimplicit-procedure flag.
A potential source of bugs is unintended integer division. Computing a value such as 1/2 will yield 0, which is most likely not your intention. The compiler can detect this and issue a warning when you add the -Winteger-division flag. However, remember that this is a compile time check, so you will not get a warnings for an expression such as a/b where a and b are integer variables, but their values are the result of a computation.
Some additional checks on implicit conversion between types and kinds can be activated using the -Wconversion-extra flag.