What do different schools of contemporary philosophy think about the Platonic Theory of forms? Is it approved by any philosophers today? What are the different thoughts towards it?
The extent to which Plato intended for his theory of forms to be taken literally is debated. If you want to be sceptical about it Plato never directly says anything himself, so definitively attributing a position to him is difficult.
The current debate is moreso centered on universals rather than forms. The concept of a universal was introduced by Aristotle in his Metaphysics. For all intents and purposes you can assume a Form is the same as a Universal.
The literal interpretation of it, what some would call hard realism, is rarely put forward seriously. Realism would posit that universals are real and exist outside our minds. A more metaphorical reading of the text might lead to a form of nominalism. Depending on the type of nominalism, the existence of universals is either accepted (under some preconditions) or denied.
For further reading consider the following: