For several years, the metastable helium triplet line has been successfully used as a tracer to probe atmospheric escape in transiting exoplanets. This absorption in the near-infrared (1083.3\,nm) can be observed from the ground using high-resolution spectroscopy, providing new constraints on the mass-loss rate and the temperature characterizing the upper atmosphere of close-in exoplanets. The aim of this work is to search for the He triplet signature in 15 transiting exoplanets ---ranging from super-Earths to ultrahot Jupiters--- observed with SPIRou a high-resolution ($R near-infrared spectropolarimeter at the CFHT, in order to bring new constraints or to improve existing ones regarding atmospheric escape through a homogeneous study. We developed a full data processing and analysis pipeline to correct for the residual telluric and stellar contributions. We then used two different 1D models based on the Parker-wind equations and nonlocal thermodynamic equilibrium (NLTE) radiative transfer to interpret the observational results. We confirm published He triplet detections for HAT-P-11\,b, HD\,189733\,b, and WASP-69\,b. We tentatively detect the signature of escaping He in HD\,209458\,b, GJ\,3470\,b, and WASP-76\,b. We report new constraints on the mass-loss rate and temperature for our three detections and set upper limits for the tentative and nondetections. We notably report improved constraints on the mass-loss rate and temperature of the escaping gas for TOI-1807\,b, and report a nondetection for the debated atmospheric escape in GJ\,1214\,b. We also conducted the first search for the He signature in GJ\,486\,b since its discovery and report a nondetection of the He triplet. Finally, we studied the impact of important model assumptions on our retrieved parameters, notably the limitations of 1D models and the influence of the H/He ratio on the derived constraints.