By a theorem of Mazur-Ogus (Katz' conjecture) the $m$-dimensional Newton polygon of a variety lies above or is equal to its $m$-dimensional Hodge polygon.

A variety is ordinary if these polygons are equal (for all $m$).

For abelian varieties the $m=1$ case suffices and you see that an abelian variety is ordinary iff it is ordinary in the usual sense.

By a theorem of Grothendieck-Katz most varieties are ordinary. This is stated more precisely also in Illusie's paper you mention. 

You should take a look at Mazur's beautiful paper on Katz' conjecture.

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183533965

If you stick to the case of curves there are many open questions (to my knowledge). For example, Mazur asks in *loc. cit.* (page 659) if all five different  possible Newton polygons arising from a smooth projective curve of genus $3$ allowed by the restraint of Poincaré duality really arise from some curve or not. I don't know if this question has been answered by now (and let me add that it might actually be answered by now). 

I can't really tell you anything else on supersingular varieties. It does seem to be a nice sport to look at "strata" in the moduli spaces of abelian varieties (=Shimura varieties). For example,  every "symmetric" Newton polygon arises from an abelian variety and the Newton polygon(s) of an abelian variety are symmetric. See http://arxiv.org/abs/math/0007201 for even more beautiful statements.

For "strata" of Shimura varieties see 

http://arxiv.org/abs/1011.3230   (Wedhorn-Viehmann)
http://arxiv.org/abs/1111.6830 (Kret)

So the moral of the story is that it's already pretty difficult to prove that certain polygons arise from geometric objects, e.g., the case of the genus $3$ curves and the Shimura variety business.