We search for signals of magnetically-induced effects in the arrival directions of ultra-high-energy cosmic rays detected at the Pierre Auger Observatory. We apply two different methods. One is a search for sets of events that show a correlation between their arrival direction and the inverse of their energy, which would be expected if they come from the same point-like source, they have the same electric charge and their deflection is relatively small and coherent. We refer to these sets of events as "multiplets". The second method, called "thrust", is a principal axis analysis aimed to detect the elongated patterns in a region of interest. We study the sensitivity of both methods using a benchmark simulation and we apply them to data in two different searches. The first search is done assuming as source candidates a list of nearby active galactic nuclei and starburst galaxies. The second is an all-sky blind search. We report the results and we find no statistically significant features. We discuss the compatibility of these results with the indications on the mass composition inferred from data of the Pierre Auger Observatory.