Is Tay al-Ard true or false? Also is there any evidence of flying carpets in the Quran or psychic powers? Also is having psychic powers haram?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

The matter of rolling up the earth during travel is narrated in some prophetic narrations. It is reported in the book of Abu Daawood, At-Tirmithi, and others May Allaah have mercy upon them, that the Prophet used to say in this supplication for travel (Allaahumma hawwin 'Alaynaa safaranaa haathaa, watwi 'Anna bu'dah). (O Allaah, facilitate our journey and let us cover it’s distance quickly.) An-Nasaa'ee in Al-Kubra and Al-Bayhaqi in his Sunan also narrated that the Prophet said encouraging night travel: “Travel at night since the earth is folded up at night”. Some scholars are of the opinion that this really happens, as Imaam Al-Munaawi said: “It means that the earth gets folded and comes near to each other. So, a traveler covers a distance at night that could not be covered during the day, especially in the last part of the night in which all acts done are blessed even more.”

Some other scholars are of the opinion that it does not happen in real, and that it is only a metaphorical expression which means that journey becomes easier and one may feel that he traveled a long way in a short time. Nevertheless, Allaah has the power to make it really fold up; nothing in this universe is beyond His Might.

As for the flying carpet, it is not mentioned clearly in the Quran but some scholars mentioned it explaining the verse: {And to Solomon [We subjected] the wind, blowing forcefully, proceeding by his command toward the land which We had blessed. …}[ Quran 21: 81].

Ibn Katheer, May Allaah have mercy upon him said: “he had a wooden carpet and used to put on it all what he needed to run his government; horses, camels, tents and soldiers, and then order the wind to take him. The wind would then go under the carpet and drive it wherever he needed, while the birds shaded him to protected him from the sun.”

As for using psychic power, probably it is a kind of magic, so a Muslim should not learn it. For details please read Fatwa 89807.