بِسْمِ اللَّهِ الرَّحْمَٰنِ الرَّحِيمِ

Common belief is that angels are in-capable of doing sin, as they only carry out orders by Allah. While it is true that angels carry out Allah’s orders however, lets see what Quran  tells us about angels having free will and whether they will be judged on the day of judgment;

2:30  وَإِذْ قَالَ رَبُّكَ لِلْمَلَائِكَةِ إِنِّي جَاعِلٌ فِي الْأَرْضِ خَلِيفَةً ۖ قَالُوا أَتَجْعَلُ فِيهَا مَن يُفْسِدُ فِيهَا وَيَسْفِكُ الدِّمَاءَ وَنَحْنُ نُسَبِّحُ بِحَمْدِكَ وَنُقَدِّسُ لَكَ ۖ قَالَ إِنِّي أَعْلَمُ مَا لَا تَعْلَمُونَ

And when your Lord said to the angels. “I will create a vicegerent on earth.” They said. “Will You place therein one who will make violence therein and shed blood?- while we do celebrate Your praises and glorify Your holy (name)?” He said. “I know what you know not.”

2:33  قَالَ يَا آدَمُ أَنبِئْهُم بِأَسْمَائِهِمْ ۖ فَلَمَّا أَنبَأَهُم بِأَسْمَائِهِمْ قَالَ أَلَمْ أَقُل لَّكُمْ إِنِّي أَعْلَمُ غَيْبَ السَّمَاوَاتِ وَالْأَرْضِ وَأَعْلَمُ مَا تُبْدُونَ وَمَا كُنتُمْ تَكْتُمُونَ

He (Allah) said. “O Adam! Tell them their names.” When he had told them, Allah said (to angels) “Did I not tell you that I know the secrets of heavens and earth, and I know what you reveal and what you conceal?”

In the above Ayas, Allah told the Malaika/angels that He knows what they reveal and what they hide. So, Malaika/angels are able to hide things in their nafs/psyche.  In Ayat 2:30, angels revealed part of what was in their nafs and concealed part of it. The part they revealed was about the potential of khalifa doing violence and bloodshed on earth. The part which they did not reveal was his ability to learn and grow and bring constant improvements in his consciousness and well being on earth. So, Allah exposed Malaika/angles for what they were hiding in their nafs.

4:172  لَّن يَسْتَنكِفَ الْمَسِيحُ أَن يَكُونَ عَبْدًا لِّلَّهِ وَلَا الْمَلَائِكَةُ الْمُقَرَّبُونَ ۚ وَمَن يَسْتَنكِفْ عَنْ عِبَادَتِهِ وَيَسْتَكْبِرْ فَسَيَحْشُرُهُمْ إِلَيْهِ جَمِيعًا

Would never the Christ disdain to be a servant of Allah, nor do the nearest angels,  If any disdains to serve him and be arrogant,-He (Allah) will gather them all together to Himself.

The Christ and the closets of the Malaika/angels do not disdain to be Allah’s servants. While the closest (to Allah) of humans and angels are humble towards Allah, any who feels embarrassed to be a servant of Allah will be the arrogant one and they all will be presented to Him. 

2:34  وَإِذْ قُلْنَا لِلْمَلَائِكَةِ اسْجُدُوا لِآدَمَ فَسَجَدُوا إِلَّا إِبْلِيسَ أَبَىٰ وَاسْتَكْبَرَ وَكَانَ مِنَ الْكَافِرِينَ

And behold, We said to the angels. “Bow down to Adam” and they all bowed down except for Iblis, he refused and was arrogant. He was from the rejecters.

The order of Allah to bow down to Adam was to Malaika/angels. They all complied, except one of them i.e Iblis. After he disobeyed and was arrogant, he became a rejecter and a Jin/shielded one

39:75  وَتَرَى الْمَلَائِكَةَ حَافِّينَ مِنْ حَوْلِ الْعَرْشِ يُسَبِّحُونَ بِحَمْدِ رَبِّهِمْ ۖ وَقُضِيَ بَيْنَهُم بِالْحَقِّ وَقِيلَ الْحَمْدُ لِلَّهِ رَبِّ الْعَالَمِينَ

And thou shall see the angels surrounding the Throne (Divine) on all sides, singing Praise to their Lord. And matter will be decided among them in Haq/Truth, and the saying will be, “Praise be to Allah, the Lord of the Worlds!”

This ayat gives an account of the judgment day, that after the judgment of people, judgment will be pronounced on angels as well.