What determines the shape-holding ability of cookies?

Cookies really only spread out because of their fat content: when it gets warm it flows, and if it flows too much before the glutens start binding to give it structure, you get flat cookie. So, in this case, if the dough is colder at the start the fat stays stable longer, and lets the cookie set up.

You can try experimenting with your fats: maybe butter instead of crisco, or vice versa. Lot of vegan types will roll in some banana to counteract the lack of eggs and animal fats. Or you could maybe add some more egg?

Things like the type of flour, gluten content, etc., make a lot of difference, in addition to fat ratios and temperature. If you're interested in the details, I'd suggest getting a copy of Bakewise, which talks a lot about exactly this issue.

There are actually numerous factors that contribute to cookie spread - not just the fat content.

(1) Sure, too much fat can contribute to spread.

(2) Cookies made with melted butter (or margarine or whatever) tend to spread more than cookies made by creaming fat with sugar first - but most modern cookie recipes seem to be using melted butter these days.

(3) Using too much white granulated sugar contributes to spread.

(4) Using pastry/cake flour (any weak flour) contributes to spread.

(5) A greased cookie sheet contributes to spread.

(6) Refrigerated cookies might spread less, but the spread is more dependent on ingredients rather than temperature of the dough.

You might consider re-posting your question based on the specific recipe or recipes that are giving you a problem. It is difficult to propose a workable solution to a general problem.

It seems to me that whole wheat flour spreads less, but I may be biased. I just don't like white flour.

I am all about freezing my cookie dough! I usually make a double batch & only bake 1-2 trays, ball up the rest in a Tupperware & throw them in the freezer. It's important to put it into balls first, or you'll have to defrost the cookie dough block & that defeats the whole purpose. I learned that the hard way! My cookies come out more plump, with a satisfying chewy (not soft, not crunchy) texture.
My friend's grandma actually ROLLS OUT the dough of her oatmeal-chocolate chip-walnut-whatever else amazing cookies, then balls them up & freezes them. I don't know if the rolling actually does anything (maybe aerates it in a unique way?) but they are the best cookies.
Sometimes I just make 2-4 cookies in the toaster oven (save energy!), then freeze the rest. Then you always have fresh, warm cookies.
Finally, the frozen dough balls are tasty on their own. Sometimes I get through those before I even bake the 2nd batch. It's especially awesome in the summer when you don't want your house to get a single degree warmer.

Definitely don't use melted butter! Softened holds together better. I think this is why freezing the dough helps them maintain their structure.

The proportion of flour and the oven heat are also related to spreading.

When I make cutout cookies, it sometimes happens that the first batch spreads a lot, because the dough was a tad too wet. But since I roll in flour/starch, not on foil, once I have to collect the cutout pieces and reroll, they keep their shape much better.

Also, if your oven is not yet up to temperature, they spread more. You really want them to start setting very quickly after the fat's melting, else they spread.