Allah created man from dust, He created the jinn from fire and He created the angels from light. It is proven by the definitive texts that it is not possible for humans to see the jinn unless they appear in a different form, such as in the form of humans and animals. The angels cannot be seen in their original form by most people unless they appear in human form. 

In the answer to question no. 70364 we have explained in detail that it is impossible for most people to see the angels in their original form, and that this was not granted to anyone of this ummah (Muslim nation) except the Messenger of Allah (peace and blessings of Allah be upon him). 

As for the angels appearing in human form, this is proven in the Quran and saheeh sunnah (authentic prophetic teachings). If an angel appears in human form, it is possible for everyone to see him and there is no difference between males and females, or old and young. But no one can be certain that a person he is seeing is an angel, because proof that they are angels is dependent on wahy (revelation) from Allah, and how can that happen after the death of the Messenger of Allah (peace and blessings of Allah be upon him)? 

The angels appeared in human form and came to Ibrahim and Loot (peace be upon them), but they did not know who they really were until the angels told them, so it is more likely that other people cannot have any knowledge of that. 

Among the evidence that the angels can appear in human form and that they can be seen in this form is the following: 

1-

Allah says (interpretation of the meaning): 

“And mention in the Book, the Quran, O Muhammad (peace and blessings of Allah be upon him) Maryam, when she withdrew in seclusion from her family to a place facing east.

She placed a screen (to screen herself) from them; then We sent to her Our Ruh [angel Jibreel], and he appeared before her in the form of a man in all respects.

She said: ‘Verily, I seek refuge with the Most Gracious (Allah) from you, if you do fear Allah.’

 (The angel) said: ‘I am only a messenger from your Lord, (to announce) to you the gift of a righteous son.’” [19:16-19] 

2-

Allah says (interpretation of the meaning): 

“Has the story reached you, of the honoured guests [three angels; Jibreel along with another two] of Ibrahim?

When they came in to him and said: ‘Salam, (peace be upon you)!’ He answered: ‘Salam, (peace be upon you ),’ and said: ‘You are a people unknown to me.’

Then he turned to his household, and brought out a roasted calf [as the property of Ibrahim (Abraham) was mainly cows].

And placed it before them (saying): ‘Will you not eat?’

Then he conceived fear of them (when they ate not). They said: ‘Fear not.’ And they gave him glad tidings of a son having knowledge (about Allah and His religion of true Monotheism).” [51:24-28] 

3-

Allah says (interpretation of the meaning): 

“And when Our messengers came to Loot, he was grieved on account of them and felt himself straitened for them (lest the town people should approach them to commit sodomy with them). He said: ‘This is a distressful day.’” [11:77] 

4-

It was narrated that ‘Umar ibn al-Khattab (may Allah be pleased with him) said: Whilst we were with the Messenger of Allah (peace and blessings of Allah be upon him) one day, a man came to us whose garment was exceedingly white and whose hair was exceedingly black, and there were no signs of travel on him, and none of us knew who he was. He came and sat before the Prophet (peace and blessings of Allah be upon him), resting his knees against his and placing his hands on his thighs. He said: O Muhammad, tell me about Islam. … He [‘Umar] said: Then he went away. I stayed there for a while, then he (the Prophet (peace and blessings of Allah be upon him)) said to me: “O ‘Umar, do you know who that questioner was?” I said: Allah and His Messenger know best. He said: “That was Jibreel, who came to you to teach you your religion.” [al-Bukhaari].

5-

The story of the trial of the leper, bald man and blind man among the Children of Israel, in which it says that Allah sent an angel in human form to test them. [al-Bukhaari and Muslim].

6.

It was narrated from Abu Hurayrah (may Allah be pleased with him) from the Prophet (peace and blessings of Allah be upon him) that “A man visited a brother of his in another town, and Allah sent an angel to wait for him on the road. When he came to him, he said: Where are you headed? He said: I am headed to a brother of mine in this town. He said: Have you done him any favour for which you hope to be recompensed? He said: No, but I love him for the sake of Allah. He said: I am a messenger from Allah to you, to tell you that Allah loves you as you love him for His sake.” [Muslim]

And there is other saheeh (authentic) evidence from which it is clear that the angels can be seen, but only in the form in which Allah commands them to appear; and then everyone that Allah decrees should see them will see them. 

The one who claims that he sees the angels in their original form is a liar or is delusional. The Prophet (peace and blessings of Allah be upon him) saw Jibreel in his true form, filling the horizon with six hundred wings. Who can claim that he saw him at that time? Thus it is clear that there is no one among this ummah who can see the angels in their true form except the Prophet (peace and blessings of Allah be upon him). As for children, they are like other people – they cannot see them unless they appear in human form. 

Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said: 

Hence humans are unable to see an angel in his true form unless Allah helps them, as He helped our Prophet (peace and blessings of Allah be upon him). Allah says (interpretation of the meaning): 

“And they say: ‘Why has not an angel been sent down to him?’ Had We sent down an angel, the matter would have been judged at once, and no respite would be granted to them.

And had We appointed him an angel, We indeed would have made him a man, and We would have certainly confused them in which they are already confused (i.e. the Message of Prophet Muhammad (peace and blessings of Allah be upon him)).” [6:8-9] 

More than one of the salaf (pious predecessors) said: They are not able to see an angel in his true form. If We sent an angel down to them, We would make him appear in human form, and in that case they would be confused as to whether he was an angel or a man. So what benefit would it be if We sent an angel to them? So We sent to them a man of their own kind, whom they could see and learn from, and this was perfect kindness and mercy to mankind. [Minhaaj al-Sunnah al-Nabawiyyah]

For more details about the angels please see the answer to question no. 843. 

And Allah knows best.