Assuming they actually are eggs in the pic - could be whiteflies or aphid shell cases - then they look like leaf miner eggs to me but there are other possibilities. Spinach has several common insect pests but leaf miners are the only one I can think of offhand that lay white eggs.

Dave

They look like really small eggs but could be shells. I get them all the time. I have not seen any evidence of the larva anywhere so likely aphid shells.

I can't tell you what the larger white clumps are (may well be eggs), but, in your first photo, you can see a spread of tiny white flecks across the leaf. They look like teeny tiny eggs but, after doing some reading myself from having a similar problem, they are in fact little oxalate crystals that the spinach naturally releases. More on this here: http://www.helpinggardenersgrow.com/2013/01/spinach-gritty-truth.html