Praise be to Allah.
Singling out a day for observing a naafil fast is permissible, unless it is a Friday or Saturday, or it is the day of ‘Ashoora which is the tenth day of Muharram, in which case it is mustahabb to fast the day before or the day after as well.
As for Sunday, Monday, Tuesday, Wednesday or Thursday, there is nothing wrong with fasting them on their own, rather it is Sunnah to fast on Mondays and Thursdays.
Al-Bukhaari (1985) and Muslim (1144) narrated that Abu Hurayrah (may Allaah be pleased with him) said: I heard the Prophet (peace and blessings of Allaah be upon him) say: “No one of you should fast on Friday, unless he fasts (a day) before it or after it.”
According to a report narrated by Muslim: “do not single out the day of Friday for fasting, unless that coincides with a fast that one habitually observes.”
Al-Nawawi (may Allaah have mercy on him) said: These ahaadeeth appear to support the view of the majority of the companions of al-Shaafa’i, which is that it is makrooh to single out Friday for fasting unless that happens to coincide with a fast that one habitually observes. If he joins it to a fast on the day before or the day after, or it coincides with a day that he habitually fasts, such as if he vowed to fast on the day that a loved one was healed for the rest of his life, and that happens to be a Friday, then it is not makrooh, because of these ahaadeeth.
The scholars said: The reason why it is forbidden is that Friday is a day of du’aa’, dhikr and worship, such as ghusl, going to the prayer early and waiting for the prayer, listening to the khutbah and reciting a lot of dhikr after it, because Allaah says: “Then when the (Jumu‘ah) Salaah (prayer) is ended, you may disperse through the land, and seek the Bounty of Allaah (by working), and remember Allaah much” [al-Jumu’ah 62:10]. And there are other acts of worship to be done on this day, so it is mustahabb not to fast then, so that one will have more energy to do these duties and perform them in an energetic manner and be happy to do them and enjoy them without feeling bored or tired. This is like the pilgrim on the day of ‘Arafah in ‘Arafah: the Sunnah is for him not to fast, as stated above.
If it is said: If that was the case, the prohibition would still apply to fasting the day before or the day after, because the reason still applies, the response to that is that he will attain reward for fasting the day before or the day after, which will make up for any shortfall in his duties on Friday that occurs because of his fasting. This is the correct view on the wisdom behind not singling out Friday for fasting.
And it was said that the reason is that there is the fear of going to extremes in venerating Friday, so that become infatuated with it as some people became infatuated about Saturday. But this is a weak view which is contradicted by the fact the Muslims offer Jumu’ah prayer and do other acts which are an expression of veneration of that day.
And it was said that the reason for this prohibition is lest it be thought that it is obligatory. This is a weak view which is contradicted by the fact that it is recommended to fast on Mondays, so no attention should be paid to this unlikely notion. Similarly it is recommended to fast on the day of ‘Arafah and the day of ‘Ashoora’ and so on. The correct view is that which we have stated above, and Allaah knows best. End quote.
As for Saturday, it is makrooh to single it out for fasting. The reason for that is that when a person fasts he becomes weak and he omits some things that he used to do, so he becomes like the Jews who do not work on Saturday. That has been discussed in the answer to question no. (the reason why it is forbidden to fast on Saturdays). 
The reason why it is forbidden to fast on Saturdays
I have read on your site that the Prophet (peace and blessings of Allaah be upon him) forbade fasting on Saturdays. What is the reason for this prohibition?
It is makrooh to single out Saturday for fasting. Some of them gave the reason as being that it is the eid of the Jews. But more important than that is the fact that it is a day on which the Jews refrain from working, and they rest on that day. If the Muslim were to fast on that day, the fast would keep him from working, and he would give up some things that he used to do because of his fast, because fasting makes one thirsty and hungry, so he would be like the Jews in not working on this day. End quote.
Fataawa al-Shaykh Muhammad ibn Ibraaheem (may Allaah be pleased with him) (4/306).
With regard to ‘Ashoora’, Muslim (1134) narrated that ‘Abd-Allaah ibn ‘Abbaas (may Allaah be pleased with him) said: When the Messenger of Allaah (peace and blessings of Allaah be upon him) fasted on the day of ‘Ashoora’ and enjoined this fast, they said: O Messenger of Allaah, it is a day that is venerated by the Jews and Christians. The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Next year – in sha Allaah – we will fast on the ninth day.” He said: But the next year the Messenger of Allaah (peace and blessings of Allaah be upon him) had passed away.
According to another report: “If I live until next year, I shall fast the ninth.”
Al-Nawawi (may Allaah have mercy on him) said: Some of the scholars said: Perhaps the reason why the ninth is to be fasted as well as the tenth is so as not to resemble the Jews by singling out the tenth. The hadeeth indicates this. And it was said that it was so as to be on the safe side and ensure that you are fasting on the tenth. But the first is more likely to be correct. And Allaah knows best. End quote.
It should be clear to you that defining the reason behind it is something concerning which the scholars exercised ijtihaad, and you have seen that al-Nawawi (may Allaah have mercy on him) regarded as mistaken the views which tried to explain why it is forbidden to single out Fridays for fasting.
Hence the Muslim should submit to the rule of sharee’ah, whether he understands the reason and wisdom behind it or not. He should accept and submit to the words of the Messenger (peace and blessings of Allaah be upon him) as Allaah says (interpretation of the meaning):
“It is not for a believer, man or woman, when Allaah and His Messenger have decreed a matter that they should have any option in their decision. And whoever disobeys Allaah and His Messenger, he has indeed strayed into a plain error”
May Allaah help us and you to do that which He loves and which pleases Him. And Allaah knows best.