There are three types of spinach (Spinacia oleracea), and they all pretty much taste the same. The big difference in them is the way that they look.
Savoy spinach has dark green crinkled leaves, grows "flatter" compared to other spinach, tends to be more cold-hardy, and becomes sweeter and crisper after a frost.
See some examples of Savoy spinach.
Flat-leaf (also called Smooth-leaf) spinach grows more upright, and because the leaves are flat, they are easier to wash.
See examples of Smooth-leaf spinach.
Semi-savoy is a hybrid between Flat-leaf and Savoy spinach, with leaves that aren't as crinkled as Savoy yet not as smooth as the Flat-leaf varieties.
A lot of times you'll find Semi-savoy spinach classified as one of the other types, depending on what it looks like.
Look at some Semi-savoy varieties of spinach.
There is a vine (Basella alba) called Malabar or New Zealand spinach, but this is not related to the spinach plant, although its leaves look and taste similar. Its claim to fame is that it grows in hot weather, which true spinach won't do.
Here are some popular Savoy types of spinach (click on the photos to learn more about each variety):
This is an heirloom savoy spinach which is very popular.
Here are some flat (or smooth) types of spinach (click on the photos to learn more about each variety):
Here I have a few of the Semi-savoy spinach type. Click on the photos to learn more about each one.
This is described as a "very heat resistant" spinach ... but from the comments it sounds as though it's heat resistant "for spinach" -- meaning you still have to plant it when it's cool and damp, just like any other spinach variety.
I hope you found these examples helpful for you to understand the various kinds of spinach. To learn a bit about how to use spinach in your edible landscape, visit this page.
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