Mild side effects are common, expected, and last 2-3 days after the vaccine is given. They can include:
Soreness or redness where the vaccine was given
Mild fever or rash a week or two after vaccination
Temporary joint aches (mostly in older teens or adults)
A very small number of children may get a minor rash and/or swelling of the glands in the neck below the ears (parotid gland) a week or so after the vaccination. This should be mild and only last a couple of days.
Serious side effects (like allergic reactions) are extremely rare - much rarer than complications from measles itself. About 1 in 1,000 children may experience a febrile seizure 6-11 days after the vaccination, however this is fewer than the number who would experience a febrile seizure if they were infected with measles, and would not stop them receiving their next dose when it’s due.