If the upper side of the leaf has yellow spots then I suspect Downey Mildew. Lots of info on the internet on the subject.

Spinach is subject to all the same fungal and bacterial diseases as most other garden vegetables. Yours looks like Septoria or Cladosporium Leaf Spot to me because of the brown spots with white centers. Septoria is the most common.

But it can be difficult to distinguish from other fungal diseases. See the link below and scroll down to the Leaf Spot section and Google pics of Septoria to compare.

Treatment is a preventative use of fungicides.

Dave

Here is a link that might be useful: Spinach Diseases