In our basic model, we study a stationary Poisson pattern of nodes on a line embedded in an independent planar Poisson field of interfering nodes. Assuming slotted Aloha and the signal-to-interference-and-noise ratio capture condition, with the usual power-law path loss model and Rayleigh fading, we explicitly evaluate several local and end-to-end performance characteristics related to nearest-neighbour packet relaying on this line. We study how these metrics depend on the density of relaying nodes and interferers, tuning of Aloha and on the external noise level. We consider natural applications of these results in a vehicular ad-hoc network, where vehicles are randomly located on a straight road. We also propose to use this model to study a "typical" route traced in a (general) planar ad-hoc network by some routing mechanism. Such a decoupling of a given route from the rest of the network in particular allows us to quantitatively evaluate the non-efficiency of long-distance routing in "pure ad-hoc" networks, previously observed in the planar scenario, and the need for a well-tuned structure of "fixed'' relaying nodes. We consider several extensions of our basic Poison-line-in-Poisson field-model, notably as Poisson-line ad-hoc network in which all nodes (including interfering ones) are randomly located on a Poisson process of lines (routes). In this case our analysis rigorously (in the sense of Palm theory) corresponds to the typical route of this network.