This will reproduce the example here.
Pull the container with Singularity and oras:
$ singularity pull oras://ghcr.io/rse-ops/singularity-mpi:mpich
You might want an allocation (with or without userns):
$ salloc --userns
Try running the container:
$ mpirun -n 6 singularity exec singularity-mpi_mpich.sif /opt/mpitest
Hello, I am rank 1/6
Hello, I am rank 2/6
Hello, I am rank 3/6
Hello, I am rank 4/6
Hello, I am rank 0/6
Hello, I am rank 5/6
And then try running with flux
$ flux start mpirun -n 6 singularity exec singularity-mpi_mpich.sif /opt/mpitest