In this paper we study the geometrical properties of the high-lying eigenfunctions (200 000 and above) which are deep in the semiclassical regime. The system we are analysing is the billiard system inside the region defined by the quadratic (complex) conformal map w=z+ lambda z² of the unit disc mod z mod <or=1 as introduced by Robnik (1983), with the shape parameter value lambda =0.15, so that the billiard is still convex and has KAM-type classical dynamics, where regular and irregular regions of classical motion coexist in the classical phase space. By inspecting 100 and by showing 36 consecutive numerically calculated eigenfunctions we reach the following conclusions: (i) Percival's (1973) conjectured classification in regular and irregular states works well: the mixed-type states 'living' on regular and irregular regions disappear in the semiclassical limit. (ii) The irregular (chaotic) states can be strongly localized due to the slow classical diffusion, but become fully extended in the semiclassical limit when the break time becomes sufficiently large with respect to the classical diffusion time. (iii) Almost all states can be clearly associated with some relevant classical object such as the invariant torus, cantorus or periodic orbits. This paper is largely qualitative but deep in the semiclassical limit and as such it is a prelude to our next paper which is quantitative and numerically massive but at about ten times lower energies.