For the quantum Ising model with ferromagnetic random couplings Ji, j > 0 and random transverse fields hi > 0 at zero temperature in finite dimensions d > 1, we consider the lowest order contributions in perturbation theory in (Ji, j/hi) to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as , where ξtyp is the typical correlation length, u is a random variable and ω coincides with the droplet exponent ωDP(D = d − 1) of the directed polymer with D = (d − 1) transverse directions. Our main conclusions are as follows. (i) Whenever ω > 0, the quantum model is governed by an infinite-disorder fixed point: there are two distinct correlation length exponents related by νtyp = (1 − ω)νav; the distribution of the local susceptibility χloc presents the power-law tail P(χloc) ∼ 1/χ1 + μloc, where μ vanishes as ξ−ωav so that the averaged local susceptibility diverges in a finite neighborhood 0 < μ < 1 before criticality (Griffiths phase); the dynamical exponent z diverges near criticality as z = d/μ ∼ ξωav. (ii) In dimensions d ⩽ 3, any infinitesimal disorder flows toward this infinite-disorder fixed point with ω(d) > 0 (for instance ω(d = 2) = 1/3 and ω(d = 3) ∼ 0.24). (iii) In finite dimensions d > 3, a finite disorder strength is necessary to flow toward the infinite-disorder fixed point with ω(d) > 0 (for instance ω(d = 4) ≃ 0.19), whereas a finite-disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension d = ∞, where ω = 0, we discuss the similarities and differences with the case of finite dimensions.