One could fast outside of Ramaḍān out of one's own will, but there are specific days in which fasting may be prohibited in certain cases. This fatwā from Sheikh ibn Bāz (رحمه الله) will explain this, ʾIn shāʾ Allāh.
He (رحمه الله) was asked:
"What are the days during which it is disliked to fast?"
Then he (رحمه الله) responded with:
"The days during which fasting is prohibited are Friday, as it is not permitted to single out Friday for fasting voluntary fasts, because the Messenger (ﷺ) prohibited that.
Likewise, Saturday should not be singled out for voluntary fasting. However, if the person fasts Friday and Saturday along with it or Thursday along with it, there is no harm in that. as the aḥādīth from the Messenger of Allāh (ﷺ) have stated. Likewise, it is prohibited to fast the day of Eid al-Fiṭr and that is (strictly) forbidden. Similarly, fasting ‘Eid-al-ʾAḍḥā and the days of at-Tashrīq should not be fasted, because the Messenger (ﷺ) prohibited that. However, concerning the days of at-Tashrīq, a report has come that proves the permissibility of fasting it to make up for the sacrifice of Hajj-al-Tamattu and Hajj-al-Qirān specifically for whoever was not able to sacrifice. This is based on what is confirmed in al-Bukhārī from ʿĀʾishah (رضي الله عنها) and ibn ʿUmar (رضي الله عنه). They both said:
'It was not permitted to fast the days of at-Tashrīq except for whoever could not make the sacrifice.' (1)
In reference to these days being fasted for voluntary fasting or for other reasons, then that is not permissible just like the day of the ‘Eid. Likewise, it is not permissible to fast the 30th day of Shaʿbān if the sighting of the new moon has not been confirmed, for verily, it is a day of doubt. It is not permissible to fast it according to the most correct view of the two opinions of the scholars. It makes no difference whether it is clear outside or cloudy due to the authentic aḥādīth that prove the prohibition of that. And Allāh is the Giver of success."
I ask Allāh to allow us to adhere to this fatwā, and that He has great mercy on Sheikh ibn Bāz and reward him for his contributions to the glorious religion of Islām.
May Allāh bless you, and shukran.
The fatwā is taken from Fatāwā Islāmīyah vol no. 3, page no. 340/341.
1. Ṣaḥīḥ al-Bukhārī 1997 & 1998