The inhalation naturally extends (arches) the spine. The exhalation naturally flexes (rounds) the spine. You can use the breath to either use this natural tendency to help you perform a desired movement better or prevent an unwanted movement. Here are two examples of how the breath can be used to help the movement:
In a Roll Up or C-Curve, the exhalation helps to flex the spine, so it makes sense to exhale to deepen the curve.
In Swan the inhalation helps you extend the spine, so breathe in while you arch your back.
Here are two examples of using the breath to prevent unwanted movement:
In Short Box: Flat Back on the Reformer or Ladder Barrel the exhalation prevents the spine from arching too much as you lean back. The inhalation will help you maintain length as you sit tall.
In Leg Pull on the mat, the exhalation prevents your lumbar spine from sagging (extension) when you lift one leg. The out-breath helps maintain a good rib to hip connection and a neutral spine.