Regardless of how your puppet is built, you'll need some ways of securing it in place while animating its individual parts. Returning to the example of a humanoid waving its hand, in order for that to look natural you'll want to move only the hand, and not the whole body; and for this, you'd need to make sure that the body is held securely in place from one frame to the next.
There are, of course, multiple ways of achieving this. One option would be to use a rigging system which hooks into the puppet's spine and allows movement of the limbs while keeping the torso firmly secured. These are great for animating jumping or running - actions which include poses where the puppet is not in direct contact with the ground. For more grounded animation, though, a simpler option is to use tie-downs in the feet.
Keeping at least one foot always secured on the ground allows you to animate individual parts of the puppet without moving its whole body. And for actions such as walking, you simply need to switch from tying down one foot to the other. It's a fairly straight-forward process which - unlike with rigs - is mostly invisible. Rigs will generally be visible in your shots, meaning that you'll then need to retouch all the affected frames in order to digitally remove the rig; but with tie-downs there will rarely be any need for this.
Of course, tie-downs systems do have their own problems. First, the armature will need to be strong enough to support the puppet, and the force you apply to it. Try to imagine the stress that is being applied to your puppet's knees when it is secured by the sole, and you are pushing on its hand...
Besides the armature, the tie-downs themselves will also need to be very strong and resilient. If you wish to secure the sole of the puppet to the ground, you will need to make sure that applying force would not cause it to break contact, or - worse yet - break off. On the other hand, though, you will at some point need to stop the connection (e.g. when walking, a foot needs to alternate between being firmly secure, and free) and the transition should ideally be as smooth as possible.
These issues can be addressed - with various degrees of success - by using one of the two most common tie-down systems: magnets or nuts-and-bolts.
Stop motion magnets can be found at crafts or specialized stores, and are rated to hold 10 KG or more (which is significantly more than what the ones on your fridge can do). One thing to keep in mind, though, is that you don't necessarily want the strongest magnets you can find; just something strong enough to account for your puppet's weight and whatever "ground" surface will be placed between the puppet and the magnet (i.e. the animation table). If the magnet is too weak, it won't hold the puppet properly; if it is too strong, you'll have a very hard time removing it, and this can possibly lead to accidental movement of the puppet. Trust me, these magnets can be very strong.
Of course, in order to be able to use magnets, you will first need to make sure that the sole of your puppet is made of a ferrous metal (so that the magnet will stick). This is very important because aluminium is not ferrous, so the typical aluminium wire armature will not be attracted by a magnet. What you'll want to do in this case is embed a ferrous piece of metal into the puppet's foot, and make sure that it is firmly attached to the armature (not the "flesh" or costume). The best thing to use for this is a steel nut; not only will this provide some grip for magnets, but it can also be used with the nuts-and-bolts system, making your puppet that much more versatile.
Nuts-and-bolts are fairly self-explanatory. A nut placed in your puppet's foot allows you secure it to the ground by having a bolt come up from beneath the ground surface and connect to it. Of course, in order to this system to work, your ground surface will need to have holes drilled into it so that the bolts can pass through. And this is where the animation table comes into play.
While you could technically animate on whatever surface you want - floor, table, etc. - it is nonetheless a good idea to treat this as a separate component worthy of your consideration. While rigs could potentially be used on any surface, tie-downs require some sort of table which is thin enough to allow the bolt or the magnet's field to pass through, and which has enough space underneath for you to stick your hand and be able to manipulate the tie-downs.
When using nuts-and-bolts, you will also need to account for the required holes - making them, and covering them up (unless you plan on doing that digitally, in post). Now drilling holes in your kitchen table is probably not the wisest thing to do, so if you're serious about creating stop motion, it really is a good idea to either buy or make a small dedicated animation table. The ideal surface for this is perforated steel, as it works great with magnets, and saves you the trouble of drilling your own holes. Plywood also works, but I'd only recommend it if you cannot find perforated steel. If you wish to make your own table, then just take a look at some existing designs, and keep in mind what you'll need when animating: the top needs to be rigid (no flexing), the feet need to be long enough to allow you to work underneath the top, and the whole thing should be stable enough so that you don't accidentally move it while shooting. A heavier table is therefore better, but even so you'll generally want to have flat feet which you can secure to whatever the animation table rests on.
