The threat that solar storms pose to our ever-modernizing society has gathered significant interest in the recent past. This is partly due to the discoveries of large peaks in the content of cosmogenic radionuclides such as radiocarbon (14C) in tree rings and beryllium-10 (10Be) and chlorine-36 (36Cl) in ice cores that were linked to extreme solar storms dated to the past millennia. To better assess the threat that they represent, we need to better quantify the relationship between their energy spectrum and their magnitude with respect to the content of the radionuclides that we measure in environmental archives such as ice cores. Here, we model the global production rate that the 59 largest particle storms coming from the Sun have induced for 10Be, 14C, and 36Cl during the past 70 years. We also consider the deposition flux in 10Be and 36Cl over the high latitudes where all Greenland ice cores are located. Our analysis shows that it is unlikely that any recent solar particle event can be detected in 10Be from ice cores. By relating these values to empirical data from ice cores, we are able to quantify different detection limits and uncertainties for 10Be and 36Cl. Due to different sensitivities to solar energetic particles, we assess that 10Be may only be suitable to detect a limited number of extreme solar storms, while 36Cl is suitable to detect any extreme particle event. This implies that the occurrence-rate estimates of extreme solar storms, based mainly on 14C and 10Be, relate to a small population of potential events.