I know there are several days aside from Ramadhan, during which Islam advises to observe fasting. I would like know what those days are.
The following are the occasions on which is suggested to fast:
Here is a hadith in Sahih Muslim that is at least one answer to your question:
He was then asked about fasting for a day and breaking on the other, whereupon he said: That is the fasting of my brother David (peace be upon him). He was then asked about fasting on Monday, whereupon he said: It was the day on which I was born. on which I was commissioned with prophethood or revelation was sent to me, (and he further) said: Three days' fasting every month and of the whole of Ramadan every year is a perpetual fast. He was asked about fasting on the day of 'Arafa (9th of Dhu'I-Hijja), whereupon he said: It expiates the sins of the preceding year and the coming year. He was asked about fasting on the day of 'Ashura (10th of Muharram), whereupon be said: It expiates the sins of the preceding year.
From http://sunnah.com/muslim/14#253 (what is quoted above is a part of the full hadith - click on the link for the full hadith)
To summarize fasts from this hadith:
- Fasting every other day is the best (the fast of Prophet Dawud)
- Fasting on Mondays
- Fasting thrice a month (13-15: the "White Days")
- Fasting on Arafah day (9 Dhul Hijjah)
- Fasting on the 10th of Muharram