CDC and the Food and Drug Administration (FDA) closely monitor the safety of vaccines approved for use in the United States. CDC uses two primary systems to monitor the safety of flu vaccines:

Vaccine Adverse Event Reporting System (VAERS): an early warning system that helps CDC and FDA monitor problems following vaccination. Anyone can report possible vaccine side effects to VAERS. Generally, VAERS reports cannot determine if an adverse event was caused by a vaccine, but these reports can help determine if further investigations are needed.
Vaccine Safety Datalink (VSD): A collaboration between CDC and nine health care organizations which allows ongoing monitoring and proactive searches of vaccine-related data.

People with egg allergies can receive any licensed, recommended age-appropriate influenza vaccine (IIV, RIV4, or LAIV4) that is otherwise appropriate. People who have a history of severe egg allergy (those who have had any symptom other than hives after exposure to egg) should be vaccinated in a medical setting, supervised by a health care provider who is able to recognize and manage severe allergic reactions. Two completely egg-free (ovalbumin-free) flu vaccine options are available: quadrivalent recombinant vaccine and quadrivalent cell-based vaccine.

Flu Vaccine Safety Information

Are flu vaccines safe?

Can I get the flu from the flu vaccine?

Do flu vaccines cause any side effects?

Are there signs or symptoms that should cause concern after getting a flu vaccine?

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What should I do if I think I am having a severe reaction to a flu vaccine?

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What should I do if I think I have been injured by the flu vaccine?

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Are there some people who should not receive a flu vaccine?

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Should pregnant women receive a flu vaccine?

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How is the safety of flu vaccines monitored?

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Source

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Are there signs or symptoms that should cause concern after getting a flu vaccine?

What should I do if I think I am having a severe reaction to a flu vaccine?

What should I do if I think I have been injured by the flu vaccine?

Are there some people who should not receive a flu vaccine?

Should pregnant women receive a flu vaccine?

How is the safety of flu vaccines monitored?

Source