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.