The most formal version of this sentence is:
- There have been many instances of outbreak in vaccinated populations.
In existential constructions like this, it is common for the verb to take its agreement from the noun phrase after the verb.
However! This is not always the case. When the auxiliary is contracted with the word there, we often see no agreement between the auxiliary verb and the noun phrase after it. This happens most often in speech and semi-formal or informal writing. In this case we have the verb BE appearing in a present perfect construction with the auxiliary verb HAVE:
- There's been many instances of outbreaks in vaccinated populations.
In the example above we see a contraction of there and the auxiliary has. It does not matter that the following noun phrase instances of outbreaks is plural, we can still use the singular form has.
Notice, however, that is not usually considered grammatical if the words there and has are separate:
- *There has been many instances of outbreaks in vaccinated populations. (X)
There are some very special cases, however, where we might be able to have a plural noun after the verb phrase has been:
- There has been one funeral, two weddings, an earthquake and a change of government since you left.
In the sentence above, we see a co-ordination of noun phrases which would normally be considered plural. However, we see a singular verb form, has in this example.
If you are doing a test at school, it is probably better to make the auxiliary verb agree with the following noun phrase. But if you want to know what native speakers actually do, and what is actually grammatical, the answer is that both there have and there's are perfectly grammatical in this sentence. We do not often find there's with a plural noun phrase in formal writing though.
Hope this is helpful!