Data from 1990-2016 for commonly administered vaccines suggests anaphylaxis occurs at a rate of 1.3 cases per 1 million doses administered. However, two cases associated with the BioNTech/Pfizer vaccine were reported within a few days of it being authorised, prompting the MHRA to update its guidance on 9 December. An article in the publication Science said a total of eight cases had occurred by 21 December and looked at the case for PEG (polyethylene glycol) being the culprit, as this is the first time PEG has been included in a vaccine.18,19.20 
The MHRA has relaxed the early contraindication around anaphylaxis: vaccines can be used unless the individual has a known hypersensitivity to any ingredient. However, anyone who experiences anaphylaxis following the first dose should not have a second. People who react to polysorbate (not a component in the Covid-19 vaccines) need to be aware they may have a cross-reaction to PEG.12,19,21 
More typical side effects include injection site tenderness/pain, headache, fatigue, myalgia, malaise, fever, arthralgia, and nausea. The AstraZeneca vaccine data suggests adverse effects after the second dose tend to be milder with fewer cases reported.8,10 
However, the US EUA for the Moderna mRNA vaccine says that more people experienced the common side effects after the second dose, “so it is important for vaccination providers and recipients to expect that there may be some side effects after either dose, but even more so after the second dose”.15 
Clearly, data about any potential longer-term adverse effects has yet to emerge, nor has there been any research on vaccine-drug interactions.8,10 

[This article is based on information available at 8 January 2021]