Randomises capitalisation of letters like bird spongebob
Unnecessary comments

You have a lot of comments telling WHAT the code does. Comments should tell -why- you do it in a certain way or what else you have tried but failed. Just repeating in comment what the code says is a waste of time (for the reader and the writer as well)

Method name

You method is called randCap. I prefer just naming it what it does, so randomCapitalize() for example. Makes the code more readable. 

Logic

I tried to understand what the logic behind the prevCharacter is. It is not immediately clear how it works, and what is has to do with the blacklist.

I think you mean to make sure there are not many lowercase or uppercase characters in a row?

You don't need such a long blacklist, because you can safely call toUpperCase() and toLowerCase() on comma's etc.

Code in main loop

You can move a lot of the code out of the main loop, by adding a method public String spongeBobify(String s) to the class. The main method will only take care of the input/ouput. 

Unnecessary calculations

What is wrong with:

boolean isCap = Math.random() &gt;= 0.5;


or, if you have a Random object, random.nextBoolean()?

Closing the scanner

You need to close your Scanner, you can do this by using a try-with-resources:

try (Scanner scanner = new Scanner(System.in)) {
    ...