Classical network-formation games are played on a directed graph. 
Players have reachability objectives: each player has to select a path from his source to target vertices. 
Each edge has a cost, shared evenly by the players using it. 
We introduce and study network-formation games with regular objectives. 
In our setting, the edges are labeled by alphabet letters and the objective of each player is a regular language over the alphabet of labels.
Unlike the case of reachability objectives, here the paths selected by the players need not be simple, thus a player may traverse some edges several times. 
Edge costs are shared by the players with the share being proportional to the number of times the edge is traversed. 
We study the existence of a pure Nash equilibrium (NE), the inefficiency of a NE compared to a social-optimum solution, and computational complexity problems in this setting.