to follow up on Bernie's post, the really heavy ones tend to be ones that are meant for non-sewing projects. If you need to stitch thru the fusing material, be sure you have the kind for sewing. The end of the bolt or attached cover should clearly state whether it is for sewing or not.
There are some that are slightly tacky fusibles which are good for holding your fabric and fusible together until you're sure it's where you want. However, as Pam says above, you might not want it if you've got black hairy dogs or cats around since it picks up dust and lint.
Paper backed fusible is good if you want to trace designs for appliques, but as stated above, some papers peel off better than others. In some cases, if the paper doesn't peel off, it may be because you haven't left it to cool long enough, or you left it too long, you may have used an iron that was either too cool or too hot, or something else entered the equation to mess it up. Find one that fits your moods and needs and stick with it.
And no matter what weight fusible you get, if you want even less stiffness, cut out the center of the fusible piece before you adhere it to the fabric so there is only fusible around the edges...just make sure to lay the piece down really flat before you iron it.
Using a teflon pressing sheet or even old paper to protect your iron from extra glue is worth the effort or expense.
Whatever you get, mark it well so you don't get confused at home. And start with small pieces and test it.