First of all, (3) is incorrect. I would say (2) is most common.
(1) is also correct but sometimes it sounds unnatural when you put some verbs using '안 -verb stem + 고 싶어요.'
Here are some examples.
* 숙제하다 (to do one's homework)
- 숙제하고 싶지 않아요. (I don't want to do my homework.) : Correct!
- 안 숙제하고 싶어요. --> Incorrect!
* 수영하다 (to swim)
- 수영하고 싶지 않아요. (I don't want to swim.) : Correct!
- 안 수영하고 싶어요. ---> Incorrect!
* 이사하다 (to move)
- 이사하고 싶지 않아요. (I don't want to move.) : Correct!
- 안 이사하고 싶어요. --> Unnatural!
* 운동하다 (to exercise)
- 운동하고 싶지 않아요. (I don't want to exercise.) : Correct!
- 안 운동하고 싶어요. --> Unnatural!
* The reason I said 'unnatural' for some sentences is because some people might say those expressions but they sound very unnatural. As you might have noticed, these verbs in sample sentences can be separated into two parts. For example, 운동하다 = 운동 (exercising) + 하다 (to do). These kind of '하다' verbs need (2) forms to make them negative sentences.
But the other kind of '하다' verbs which cannot separated into two parts, such as 미워하다 , 좋아하다, 싫어하다 can use both (1) and (2) to make negations.
e.g.) 안 미워하고 싶어요. or 미워하고 싶지 않아요. ( Both OK )
안 좋아하고 싶어요. or 좋아하고 싶지 않아요. ( Both works!)
안 싫어하고 싶어요. or 싫어하고 싶지 않아요. ( Both fine!)
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