These days, we see large amount of money spent on building fancy mosques. Is this permissible?

Although we tend to look at the fancy masjids with all colors, carvings and calligraphy as something really beautiful, Prophet Muhammad (ﷺ) said that one of the signs of last days would be the beautification of masjids.

It was narrated that Anas bin Malik said: “The Messenger of Allah (ﷺ) said: 'The Hour will not begin until the people compete in (building) mosques.'"[1]

The purpose of masjid is not its beautification; it is for prayer and remembering Allah.  Aisha (رضى الله عنها) , the wife of the Prophet (ﷺ), said, "Abu Jahm ibn Hudhayfa gave the Messenger of Allah a fine striped garment from Syria and he did the prayer in it. When he had finished, he said, 'Give this garment back to Abu Jahm. I looked at its stripes in the prayer and they almost distracted me.'” [2]

Therefore, even the rugs with different designs and patterns on it are actually counter-productive, because when you stand in prayer, these lines on the rug distract you. The rug should, therefore, be plain. That is closer to the Sunnah.

References: 

[1] Sunan Ibn Maajah: 739

[2] Muwatta Malik, Book 3, Hadith 72