Firstly: Defining Istijmar
Istijmar linguistically: taken from jamarat and jimar, which are small rocks.
Istijmar technically: removing filth from the exit using stones or their like.
Secondly: The Ruling of Istijmar
It is permissible to use stones for removing najasah from the exit, and this is the summarised view. Consensus has been quoted over the matter by: Ibn Hazm, Ibn `Abd al-Barr, Ibn Rushd, Ibn Taymiyyah, and Ibn Muflih.  Difference of opinion took place when considering the woman’s urine and sing stones when water is available.
Thirdly: What is Used for Istijmar
It is permissible to use any pure, cleaning substance with which the filth can be removed, like stones, tissues, and papers and parchments that carry no sanctity. It is not a condition to solely use stones.
Fourthly: The Obligations and Conditions of Istijmar That it is with three stones
Istijmar with less than three stones does not suffice.  Or one stone that has three distinct parts, as will follow. This is the position of the Shafi`is, Hanbalis, the people of Hadith. It is the choice of Ibn al-Mundhir, Abu al-Faraj from the Malikis, Ibn Hazm, Ibn Taymiyyah, and Ibn Baz.
- The ruling of using one stone three times when wiping in performing istijmar
It suffices to use one stone that has three separate parts three times, once for each wipe. This is the position of Shafi`is and Hanbalis, and it is the verdict of Ishaq and Abu Thawr, as well as the choice of Ibn `Uthaymin. That the stones are pure
The stones must be pure as a condition for using them. This is the position of the majority: Malikis, Shafi`is, and Hanbalis. That it is cleaning
The stones, or what is used in their place, must be able to adequately clean.  Ibn `Uthaymin said: “That which does not clean [would not do so] because of its smoothness, such that it is too smooth, or damp, like a damp rock, or some damp soil paste; or if the exit part has dried, and this is if the stone is able to clean but the exit itself can no longer be [easily] cleaned.” Al-Sharh al-Mumti` (134/1) This is by agreement of the four schools of jurisprudence. That it is not fluid
Whatever is used for istijmar cannot be fluid as a condition, and this is the position of the majority: Malikis, Shafi`is, and Hanbalis. That it is not bone or dung
Using bone or dung does not suffice as a substance for istijmar. This is the position of Shafi`is, Hanbalis, the choice of Ibn al-Mundhir, Ibn Hazm, Ibn `Abd al- Barr, Ibn Baz, Ibn `Uthaymin, and the majority of scholars. That it is not venerated
Whatever is used for istijmar must not be a thing that is venerated and respected, like paper with the mention of Allah on it, or food, and things of that nature. This is by agreement of the four schools of jurisprudence.
Fifthly: Using an Odd Number of Stones
It is mustahabb to use an odd number of stones for istijmar, such that if three stones do not suffice for cleanliness, one would add two to make the total five, or four to make the total seven, and so on. This is the position of Shafi`is, Hanbalis, the choice of Ibn Taymiyyah, al-San`ani, al-Shawkani, Ibn Baz, and Ibn `Uthaymin.
Sixthly: Traces Left after Istijmar
Traces left after properly performing istijmar are pardoned and overlooked, and this is by agreement of the four schools of jurisprudence.