Measles virus undergoes error-prone replication like other RNA viruses, but over time, it has remained antigenically monotypic. The constraints on the virus that prevent the emergence of antigenic variants are unclear. As a first step in understanding this question, we subjected the measles virus genome to unbiased insertional mutagenesis, and viruses that could tolerate insertions were rescued. Only insertions in the nucleoprotein, phosphoprotein, matrix protein, as well as intergenic regions were easily recoverable. Insertions in the glycoproteins of measles virus were severely under-represented in our screen. Host immunity depends on developing neutralizing antibodies to the hemagglutinin and fusion glycoproteins; our analysis suggests that these proteins occupy very little evolutionary space and therefore have difficulty changing in the face of selective pressures. We propose that the inelasticity of these proteins prevents the sequence variation required to escape antibody neutralization in the host, allowing for long-lived immunity after infection with the virus. Display omitted •An insertional mutagenesis screen of the measles virus genome is performed•Some genomic regions are tolerant of insertions, but the F or H glycoproteins are not•In comparison to other viruses, the glycoproteins of measles are highly inflexible•The inflexibility of measles F and H may contribute to measles antigenic stability It is unclear why measles virus is antigenically monotypic. Fulton et al. perform a viral mutagenesis screen and find that the glycoproteins (the major immune response targets) are exceptionally inflexible. These data suggest that the intrinsic rigidity of the glycoproteins is an important constraint on the development of antigenic diversity.