Heat the olive oil in a large pot over medium heat. Cook the onion and garlic in the hot oil until softened, 5 to 7 minutes. Stir the carrot and celery into the mixture and continue cooking until the carrot and celery have also softened, another 7 to 10 minutes.
Pour the chicken broth into the pot and bring to a boil; cook at a boil for 5 minutes. Add the mussels; cover the pot and allow the mussels to simmer in the broth mixture just until they begin to open, about 5 minutes. Discard any mussels which do not open. Stir the spinach into the broth just before serving hot.
