Smooth maps having only singularities with Boardman symbol (1,0)

We will determine the homotopy type of the subspace Ω1,0(n,p) of J2(n,p) consisting of all 2-jets of smooth map germs, which are regular or have singularities with Boardman symbol (1,0). This is aimed at classifying Ω1,0-regular maps, namely smooth maps having only such singularities by the homotopy principle due to [A. du Plessis, Comment. Math. Helv. 50 (1975) 363] and [M. Gromov, Math. USSR Izv. 3 (1969) 671]. As applications we will give conditions for the existence and non-existence of an Ω1,0-regular map.