Finding the perfect set of skis can easily be compared to trying to find the perfect relationship, in short it's complicated! It's also something that's very much subjective and can be a lot of hassle, but once you've found the right 'partner'all that effort will be worth it. Be careful not to get too bogged down in this relationship/ski analogy, as while in a relationship they say that opposites attract, in the case of you and your skis, you should probably share a similar goal. For example, it wouldn't make much sense to trek into the backcountry powder on a skinny set of park skis. It wouldn't be fun and no doubt you'd soon be pretty frustrated. Such a relationship is almost certainly doomed to failure. But as we said, it is all very subjective, so please bear in mind that nothing is gospel.
Firstly you should be aware that skis can be regular or twin tip. It's pretty self explanatory, but twin tip basically means that the ski has two tips because unlike regular skis the tail is also curved upwards. This is designed to help you easily ski in both directions, ie ride regular and backwards (otherwise known as switch). For freestylers twin tips are pretty much a no brainer.
Concepts such as zero-camber, negative camber and rocker shapes resound throughout the freeski world. These terms might sound quite perplexing, but they are just used to describe the shape of the ski when you view it from the side. Until recently, ski manufacturers were all sticking to producing traditional camber skis and these still exist, but now you can also buy skis that are the reverse i.e. negative camber. This shape works really well in the powder, but can bring disadvantages on the slopes and in the parks, which is why many ski manufacturers have gone on to create hybrids, which are skis that combine positive, negative and zero camber in various extremes, designed to give you the best of both worlds.
Determining the proper flex in freeski models is not as easy as it used to be with a standard set of old school entry level skis. It's simply no longer a question of "hard" or "soft". With all the new shapes, it is impossible to isolate the flex. Therefore, we'll keep this brief. In general the stiffer the ski the more responsive it will be, however stiffer skis have the downfall of being less forgiving, which is something to bear in mind. A softer flex on a set of wide skis will love the backcountry jibbing and powder surfing. Freestyle skis also prefer a softer flex, as it helps to absorb shocks and be more forgiving on landings. Be aware that if you go too soft then you'll loose some pop, so you might not be able to pop off those rails quite as you'd expeect. But really flex is very much a matter of personal taste!
Narrower waist widths are quicker edge to edge when turning, whilst wider waist widths will help flotation in powder and chopped up snow. In freeriding you can generally apply the thesis: "The wider the ski, the better it is in powder." and most freeskiers wouldn't contradict this. But the trouble comes in that there are now so many forms of "wide". Waists of 85 mm or even 90 mm brings with it the stability that is needed for monstrous obstacles, which when you see kickers the size of a block of flats with the run in reminiscent of a long-range missile, it's not surprising that even park skis have now become a little wider. Die hard rail riders will find narrower skis more agile.
In terms of the ski's construction, there are two main types - cap and sandwich. Cap construction involves laying fiberglass or similar over the top layer of the skis The fiberglass extends over the core from one edge to the other. This type of ski has the advantage of being very light. Sandwich construction is pretty much like a ski sandwich. Layers are stacked on top of each other and bound together. The usual order goes base layer, then fiberglass, then metal, then the core, then another layer of fiberglass, and lastly the topsheet. The result of this is a vertical sidewall which will help provide plenty of edge grip.