Let \(X\subset \mathbb{P}(V)\) be a non-degenerate smooth projective variety. The \(k\)-th osculating space to \(X\) at \(x \in X\) is spanned by the \(k\)-jets of sections in \(V\) evaluated at \(x\). The paper under review is devoted to the study of the \(k\)-th osculatory behavior of smooth projective surfaces fibered in conics. The special structure of these conic fibrations is reflected in the osculatory behavior for \(k \geq 3\). In particular, for \(k=3\), they can be isomorphically projected in \(\mathbb{P}^8\) without affecting the general dimension of the \(3\)-osculating spaces. An interesting result (see Cor. 15) is that a conic fibration in \(\mathbb{P}^8\) is either hypo-oculating (the dimension of the general \(3\)-osculating spaces is smaller than expected) or it has a \(1\)-dimensional inflectional locus \(\Phi_3(X)\) (where the osculating spaces has dimension smaller than general). The presence of curves of low degree transverse to the fibers is of importance to study the dimension of the osculating spaces. An explicit description of the inflectional loci (when of expected codimension) in terms of Chern classes is stated. Several relevant examples are also provided (Castelnuovo surfaces, \(3\)-osculatory behavior of Del Pezzo surfaces).