The screen does trigger the collapse of the particle wave function to a specific point on the screen. The probability of collapse at a specific point is given by the magnitude squared of the wave function at the screen. Thus, when one carries experiment many times (shooting one particle after another), an interference pattern appears, in the form of darker areas where more particles have collapsed, and lighter areas where the probability of finding a particle is low. 

It is for this reason that the descriptions of the double slit experiment always talk about *a beam of particles*, going through the trouble of explaining that the particles are shot one after another, so that we can neglect the interaction between them. Although a single particle is sufficient for setting the Schrödinger equation and calculating the interference pattern, it is not in practice sufficient for observing the interference. This is consistent with the canonical formulation of quantum mechanics, where the measurement is always done repeatedly on many objects, *prepared in the identical quantum state*.

However, as long as the place, where the collapse occurs on the screen, does not allow us to distinguish through which slit the particle has passed, the interference fringes are preserved.