Amyotrophic lateral sclerosis (ALS) involves the rapid degeneration of upper and lower motor neurons leading to weakening and paralysis of voluntary movements. Mutations in copper‐zinc superoxide dismutase 1 (SOD1) are a known genetic cause of ALS, and the SOD1 G93A mouse has been used extensively to investigate molecular mechanisms in ALS. In recent years, evidence suggests that ALS and frontotemporal dementia form a spectrum disorder ranging from motor to cognitive dysfunctions. Thus, we tested male and female SOD1 G93A mice for the first time before the onset of debilitating motor impairments in behavioural domains relevant to both ALS and frontotemporal dementia. SOD1 G93A males displayed reduced locomotion, exploration and increased anxiety‐like behaviours compared with control males. Intermediate‐term spatial memory was impaired in SOD1 G93A females, whereas long‐term spatial memory deficits as well as lower acoustic startle response, and prepulse inhibition were identified in SOD1 G93A mice of both sexes compared with respective controls. Interestingly, SOD1 G93A males exhibited an increased conditioned cue freezing response. Nosing behaviours were also elevated in both male and female SOD1 G93A when assessed in social paradigms. In conclusion, SOD1 G93A mice exhibit a variety of sex‐specific behavioural deficits beyond motor impairments supporting the notion of an ALS‐frontotemporal spectrum disorder. Thus, SOD1 G93A mice may represent a useful model to test the efficacy of therapeutic interventions on clinical symptoms in addition to declining motor abilities.