  Surfaces of general type with positive second Segre number $s_2:=c_1^2-c_2>0$
are known by results of Bogomolov to be quasi-hyperbolic i.e. with finitely
many rational and elliptic curves. These results were extended by McQuillan in
his proof of the Green-Griffiths conjecture for entire curves on such surfaces.
In this work, we study hyperbolic properties of minimal surfaces of general
type with minimal $c_1^2$, known as Horikawa surfaces. In principle these
surfaces should be the most difficult case for the above conjecture as
illustrate the quintic surfaces in $\bP^3$. Using orbifold techniques, we
exhibit infinitely many irreducible components of the moduli of Horikawa
surfaces whose very generic member has no rational curves or even is
algebraically hyperbolic. Moreover, we construct explicit examples of
algebraically hyperbolic and (quasi-)hyperbolic orbifold Horikawa surfaces.
