Ideally you need all of them, and probably not just one hand saw. In this article we'll which ones can be skipped in a non-ideal situation (to be read "low budget") and which one is for what.
The circular saw is much necessary if you are going to build a shed from scratch and will cut all the wood yourself. It's fast and makes clean straight cuts. And I am not talking about a table-mounted circular saw although they are nice. Building a shed is not that precise woodwogking job, you are not a cabinet builder here. So a hand circular saw will do the work. Do get one, it will save you a lot of efforts to cut large boards and beams by hand. It's also much easier to keep cuts straight.
Circular saws used to be quite expensive but no more. You can buy a decent one for much less than $100. If this is over your budget, buy used one at auction for less than $50. You'll thank yoruself for this purchase during the whole project, and you can use the saw for other projects in the future.
Many home builders already have a small jig saw in their workshop. Jig saws are cheap, safe and easy to use by everyone. A decent jig saw can be bought for $20 so that's close to free. These saws however aren't very good for the work. First, it's hard to make a straight cut especially on a thicker board. It's impossible to cut a beam anything larger than 2" simply because the blade of these saws is not long enough. Finally, because the blade is thin and held only on one side the cuts are often vertically skewed.
In short, a jig saw can help you to cut a piece of plywood here and there and save you some time. But if you have a circular saw, use that instead. You may even prefer hand saws over to jig saw for such project. If you don't have one yet, don't buy it entirely for this project. Better save the money for something else.
Don't understimate the power and comfort of using a hand saw. A project like shed will require at least one large woodworking hand saw for rough cuts. While you can replace it by using a circular saw or a jig saw (less useful), the hand saw is easier to fit in areas where the other saws won't. It doesn't need electricity and doesn't break easily. And it's much safer although also carries some risk of injury.
Besides the large hand saw for rough cuts you may want to have two other saws: a dovetail saw and a keyhole saw. The dovetail saw may come handy for a joint here and there. You probably won't be using dovetails in shed building but some details on the doors, windows and the roofs will need more precise cuts than the rough saws can do. The circular also can't replace the dovetail saw - it's not good for small cuts. Do buy at least one dovetail saw - you are going to use it for years after that as well.
The keyhole saw may not be needed at all. But when you need it, you'll need it badly! Nothing else can help you cut a small hole or a vent, or to reach a misplaced piece of wood in already joint area. A jig saw will sometimes do but only if you can reach the required piece with it.
If you are not a professional woodworker don't buy expensive saws. A high-quality dovetail saw can cost over $100. You don't need that for a shed. Each of the three hand saws can be bought for under $10 with OK quality. So even if you buy all the 3 and a circular saw your budget for the most important tools for your shed building - the saws, will be under $100. Don't forget you are going to use them on all your future projects as well.