Since the beginning of the COVID-19 pandemic, vaccines have been developed to mitigate the spread of SARS-CoV-2, the virus that causes COVID-19. These vaccines have been effective in reducing the rate and severity of COVID-19 infection but also have been associated with various adverse events (AEs). In this study, data from the Vaccine Adverse Event Reporting System (VAERS) was queried and analyzed via the Cov19VaxKB vaccine safety statistical analysis tool to identify statistically significant (i.e., enriched) AEs for the three currently FDA-authorized or approved COVID-19 vaccines. An ontology-based classification and literature review were conducted for these enriched AEs. Using VAERS data as of 31 December 2021, 96 AEs were found to be statistically significantly associated with the Pfizer-BioNTech, Moderna, and/or Janssen COVID-19 vaccines. The Janssen COVID-19 vaccine had a higher crude reporting rate of AEs compared to the Moderna and Pfizer COVID-19 vaccines. Females appeared to have a higher case report frequency for top adverse events compared to males. Using the Ontology of Adverse Event (OAE), these 96 adverse events were classified to different categories such as behavioral and neurological AEs, cardiovascular AEs, female reproductive system AEs, and immune system AEs. Further statistical comparison between different ages, doses, and sexes was also performed for three notable AEs: myocarditis, GBS, and thrombosis. The Pfizer vaccine was found to have a closer association with myocarditis than the other two COVID-19 vaccines in VAERS, while the Janssen vaccine was more likely to be associated with thrombosis and GBS AEs. To support standard AE representation and study, we have also modeled and classified the newly identified thrombosis with thrombocytopenia syndrome (TTS) AE and its subclasses in the OAE by incorporating the Brighton Collaboration definition. Notably, severe COVID-19 vaccine AEs (including myocarditis, GBS, and TTS) rarely occur in comparison to the large number of COVID-19 vaccinations administered in the United States, affirming the overall safety of these COVID-19 vaccines.