In recent years, there has been a vast amount of debates.
or
In recent years, there have been a vast amount of debates.
In recent years, there has been a vast amount of debates.
or
In recent years, there have been a vast amount of debates.
The word amount exploits the common English practice of treating a count noun, which would ordinarily be plural, as a singular mass noun when it serves some rhetorical purpose. To make the sentence sound right, you need to say:
In recent years, there has been a vast amount of debate.
Making debate singular here turns it into a mass noun, agreeing with amount. The singular noun amount agrees with the singular verb has.
With this phrasing, the listener tends to imagine "debate" as a non-stop process, extending over several years. This is a common technique: mass nouns encourage people to think of the things referred to as continuously divisible substances like water, mud, or sand. The listener is led to imagine people droning on and on in "debate" (singular, as a mass noun), rather than a lot of separate debates (plural, as a count noun). This phrasing would be a good choice if you wanted to suggest that over the last few years, there have been two sides stuck in a long-running disagreement, mostly just repeating themselves or falling into ever-more complicated technicalities as they attempt to persuade.
On the other hand, if you wanted the listener to think of an overwhelming quantity of separate debates, then you need to replace or omit the word amount:
In recent years, there have been a great many debates.
In recent years, there have been a great number of debates.
In the second of these sentences, you could use the verb has (singular), agreeing with number, but when the subject is number, English lets the verb agree with the object of the preposition (or with the prepositional phrase as a whole). The plural have is better rhetoric if you're trying to emphasize a large quantity of separate debates on separate occasions. The count-noun phrasing would be a good choice if you wanted to suggest that many different ideas have been proposed during the last several years.
You should use has here.
Its number matches with amount (both are singular), which is the relevant noun to consider here. To match the verb’s number with debates (i.e. to use have) would be a mistake, since debates is the object of a prepositional phrase.
The two alternatives for your sentences are
There has been a vast amount (a lot of one occurrence)
There have been vast amounts (a lot of different occurrences)
where the number (amount/amounts) matches the verb (has/have)
For example,
There has been a vast amount of rain today in the metro area
a lot of rain fell todayThere have been vast amounts of rain today in the metro area
a lot of rain has fallen, across different areas