In the Commons the Committee of Selection chooses which MPs will serve on most of the committees of the House, including all public and private bill committees and delegated legislation committees. Their nominations must be approved by the Commons as a whole, and may be challenged.
The membership is chosen on a Wednesday and published on a Thursday in Votes and Proceedings
The Committee of Selection in the House of Lords has a similar role but only selects members for Private Bill Committees.
Further information on this subject can be found from the following link.
Commons business papers: Votes and Proceedings