Fasting in Sha’ban
It is important to note that there is a difference between:
fasting most of the month of Sha’ban, or fasting only in its first half, or fasting some days in both of its halves. This is permissible, and it is even recommended according to the majority of the madhabs. This is based on the hadith:
ولم أره صائما من شهر قط أكثر من صيامه من شعبان كان يصوم شعبان كله كان يصوم شعبان إلا قليلا
I never saw the Prophet ﷺ observing voluntary fasts more in any other month than that of Sha'ban. It appeared as if he observed fast throughout the whole of Sha'ban except a few days.
— Muslim
fasting on the 30th day of Sha’ban on which there exists doubt regarding whether it is the first of Ramadan instead. This could be because of cloudy weather or because not enough reliable witnesses could be found because of which Ramadan could not be confirmed. The ruling on this is disagreed upon. Most madhabs consider fasting on this day especially to be makruh and some consider it haram and invalid to fast on this day, based on the ahadith:
لا يتقدمن أحدكم رمضان بصوم يوم أو يومين، إلا أن يكون رجل كان يصوم صومه فليصم ذلك اليوم
None of you should fast a day or two before the month of Ramadan unless he has the habit of voluntary fasting (and if this coincides with that day) then he can fast that day.
— Bukhari For example if a person follows a pattern whereby he has a habit of fasting on every Thursday then it is permissible for him to continue this pattern on the day of doubt if falls on a Thursday.
fasting only in the second half of the month while not fasting in the rest of the month. There is disagreement on fasting in the second half of the month:
It is permissible according to the majority. They consider the hadith against it to be weak. Some have also relied on the hadith of fasting in سرر شعبان which some scholars interpret as its mid.
It is haram according to the Shafi'is and some Hanbalis have considered it makruh. They rely on the hadith:
إذا انتصف شعبان فلا تصوموا
When a half of Sha'ban remains then do not fast.
— Jami at-Tirmidhi, Abu Dawud Ibn Majah
The reconciliation with the ahadith about fasting in Sha’ban is that the permission applies to one who also fasts in the first half of the month, or one who is fasting according to some pattern and not just fasting in the last half of specifically, or one who is making an obligatory fast such as for a vow or kaffarah.
For details see: https://islamqa.info/en/answers/13726/fasting-in-the-second-half-of-shaban
Fasting on Saturday:
Fasting on Saturday is obligatory if it is an obligatory fast such as the Saturdays that fall in the month of Ramadan, or the Saturdays that fall in the 60 day period of consecutive fasting for Kaffarah.
As for voluntary fasts: there is difference of opinion: Fasting on Saturday alone (i.e. not fasting on Friday or Sunday with it) is makruh according to the Hanafis, Malikis, Shafi'is and the dominant position of the Hanbalis. This is based on the hadith:
لا تصوموا يوم السبت إلا فيما افترض عليكم وإن لم يجد أحدكم إلا لحاء عنبة أو عود شجرة فليمضغه
Do not fast on Saturday except what has been made obligatory on you; and if one of you can get nothing but a grape skin or a piece of wood from a tree, he should chew it.
— Abu Dawud, Jami at-Tirmidhi
And also based on the fact that Saturday is venerated by the Jews as the Sabbath and fasting on it is a form of imitation by venerating the day. The Hanafis have further stated that if the person who fasts on Saturday intends to imitate the Jews then it is makruh tahrimi (which is nearly equivalent to haram).
The Hanafis, Hanbalis and Shafi'is have stated that if a person follows some pattern in voluntary fasting and the fast happens to coincide occasionally with a Saturday then it is not makruh to fast on it. For example if a person fasts every alternate day then it will inevitably fall on Saturday in some weeks and that will be permissible.
Similarly, they have stated it is permissible if a person fasts on both Friday and Saturday together.
As for fasting on both Saturday and Sunday together, the Hanafis have two opinions on it. Some have held that it is makruh while others have held that it is not makruh. Ibn Abideen has preferred the second one. The Shafi'is also allow it. The reason is that neither the Jews nor the Christians venerate both of these days together and because this is analogous to fasting on Friday and Saturday together, or Fasting on Sunday and Monday together which is permissible.
Fasting on Sunday:
The Hanafis and Shafi'is have explicitly stated that fasting on Sunday alone is also makruh as it is analogous to Saturdays since it is venerated by the Christians. The Hanbalis have stated in general that fasting on the days that are venerated by the disbelievers is makruh.