Therefore celebrate the praises of thy Lord, and be of those who prostrate themselves in adoration
(Qur'an 15:98)

The Shi'ah Muslims prefer to prostrate on a small block of earth, called a Turbah, which is usually made from clay from the land of Karbala in Iraq.

According to the Shi'ah Ja'fari fiqh - which is one of the five main schools of law in Islam - prostration must be performed on pure earth or what grows on it, provided that it is not eaten or worn. This includes dust, stone, sand and grass, provided that it is not a mineral. Prostration on paper is permitted, because it is made of a material which grows on earth, but not cloth or carpets.

The legal scholars of all the Sunni schools of law concur regarding the validity of prostration on earth and that which grows on it.

Did the Prophet (s) and his Companions ever do this?

Praying on the earth was certainly the practice of the Prophet (s) and those around him.

• Narrated Abu Sa'id al-Khudri: I saw Allah's Apostle prostrating in mud and water and saw the mark of mud on his forehead.
  [Al-Bukhari, Sahih (English translation), vol. 1, book 12, no. 798; vol. 3, book 33, no. 244]

• Narrated Anas bin Malik: We used to pray with the Prophet in scorching heat, and if someone of us could not put his face on the earth (because of the heat) then he would spread his clothes and prostrate over them.
  [Al-Bukhari, Sahih (English translation), vol. 2, book 22, no. 299]

According to this hadith only in exceptional circumstances would the Prophet (s) and his Companions prostrate on cloth.

The Prophet (s) also used to have a Khumra on which he would put his forehead for prostration.

• Narrated Maymuna: Allah's Apostle used to pray on a Khumra.
  [Al-Bukhari, Sahih (English translation), vol. 1, book 8, no. 378]

• According to al-Shawkani, a famous Sunni scholar, more than ten Companions of the Prophet (s) have narrated traditions mentioning his prostration on a Khumrah.  And he lists all the Sunni sources recording these traditions which include Sahih Muslim, Sahih al-Tirmidhi, Sunan Abu Dawud, Sunan al-Nasa'i and many others.
  [Al-Shawkani, Nayl al-Awtar , Chapter of Prostration on the Khumrah, vol. 2, p. 128]

So what is a khumrah ?

• a small mat sufficient just for the face and the hands while prostrating during prayers.
  [Al-Bukhari, Sahih (English translation), vol. 1, book 8, no. 376 (as explained by the translator in paranthesis)]

Ibn al-'Athir, another famous Sunni scholar, in his Jami al-'Usul has written:

• "Khumra is [like that] upon which the Shi'ah of our time perform their prostrations."
  [Ibn al-'Athir, Jami' al-Usul, (Cairo, 1969), vol. 5, p. 467]

• "Khumra is a small mat made from palm fibres or other material…. and it is like that which the Shi'ah use for prostration."
  [Talkhis al-Sihah, p. 81]

But why the earth of Karbala?

The special characteristics of the soil of Karbala (Iraq) were known and it was an object of special attention during the time of the Prophet (s) as well as in later times:

• Umm Salama says: I saw Husayn (a) sitting in the lap of his grandfather, the Prophet (s), who had a red block of soil in his hand. The Prophet (s) was kissing the dust and weeping. I asked him what that soil was. The Prophet (s) said: "Gabriel has informed me that my son, this Husayn, will be murdered in Iraq. He has brought this earth for me from that land. I am weeping for the suffering that will befall my Husayn." Then the Prophet (s) handed the dust to Umm Salama and said to her: "When you see this soil turn into blood, you will know that my Husayn has been slaughtered." Umm Salama kept the soil in a bottle and kept watch over it until she saw on the day of Ashura, 10th of Muharram 61 A.H., that it turned to blood. Then she knew that Husayn bin Ali (a) had been martyred.

[al-Hakim, al-Mustadrak, vol. 4, p. 398]
[al-Dhahabi, Siyar a`lam al-nubala', vol. 3, p. 194]
[Ibn Kathir, al-Bidayah wa'l-nihayah, vol. 6, p. 230]
[al-Suyuti, Khasa'is al-kubra, vol. 2, p. 450; Jam` al-Jawami, vol. 1, p. 26]
[Ibn Hajar al-Asqalani, Tahdhib al-tahdhib , vol. 2, p. 346]

• 'Ali ibn Abi Talib, passed by Karbala after the battle of Siffin. He took a handful of its soil and exclaimed: 'Ah, ah, on this spot some men will be slain, and will enter Paradise without reckoning!'
  [Ibn Hajar al-Asqalani, Tahdhib al-tahdhib , vol. 2, p. 348]

Why is it compulsory to prostrate on the clay from Karbala?

It is not! But the Shi'ah prefer to prostrate on the earth of Karbala because of the importance given to it by the Prophet (s) and the Imams from his Family (Ahl al-Bayt). After the martyrdom of Imam Husayn (a), his son Imam Zayn al-'Abidin (a) picked some up, declared it to be sacred dust, and kept it in a bag. The Imams (a) used to perform prostrations on it and make a tasbih out of it, and recited Allah's praises on it.
[Ibn Shahrashub, al-Manaqib,  vol. 2, p. 251]

They also encouraged the Shi'ah to perform prostrations on them, with the understanding that it was not compulsory, but with a view to achieving greater recompense. The Imams (a) insisted that prostration before Allah must be on clean earth only and that it was preferable if it was performed on that earth of Karbala.
[al-Tusi, Misbah al-Mutahajjad,  p. 511]
[al-Saduq, Man la yahduruhu'l faqih,  vol. 1, p. 174]

The Shi'ah for a long time have kept this earth with them. Then, fearing that it might be desecrated, they kneaded it into small tablets or pieces, which are now called mohr or Turbah. During prayers we prostrate on it not as a compulsory act but in view of its special nature. Otherwise, when we have no pure soil with us, we prostrate on clean earth, or something that originates from it.

It is a pity that some people maliciously insist that the Shi'ah worship stones or that they worship Husayn (a). The truth is that we worship Allah alone by prostrating on the Turbah, not to it.  And we never worship Imam Husayn, Imam Ali, or the Prophet Muhammad (s). We worship only Allah, and it is in accordance with Allah's order that we perform prostration only on pure earth.

Conclusion:

This is the reason why Shi'ah Muslims carry small tablets, usually made from the earth of Karbala, which enables them to do prostration on this highly recommended object and to follow the sunnah of the Prophet (s).