Title 8 Chapter I Subchapter B Part 287 § 287.5
It's long, but it outlines who is authorized to question suspected aliens, who is authorized to arrest suspected aliens, who can patrol the border, etc. Here's one section:
§ 287.5 Exercise of power by immigration officers.
(c) Power and authority to arrest.
(1) Arrests of aliens under section 287(a)(2) of the Act for immigration violations. The following immigration officers who have successfully completed basic immigration law enforcement training are hereby authorized and designated to exercise the arrest power conferred by section 287(a)(2) of the Act and in accordance with 8 CFR 287.8(c): (see below)
(i) Border patrol agents;
(ii) Air and marine agents;
(iii) Special agents;
(iv) Deportation officers;
(v) CBP officers;
(vi) Immigration enforcement agents;
(vii) Supervisory and managerial personnel who are responsible for supervising the activities of those officers listed in this paragraph; and
(viii) Immigration officers who need the authority to arrest aliens under section 287(a)(2) of the Act in order to effectively accomplish their individual missions and who are designated, individually or as a class, by the Commissioner of CBP, the Assistant Secretary/Director of ICE, or the Director of the USCIS.
(referenced above)
Title 8 Chapter I Subchapter B Part 287 § 287.8
(c) Conduct of arrests —
(1) Authority. Only designated immigration officers are authorized to make an arrest. The list of designated immigration officers may vary depending on the type of arrest as listed in § 287.5(c)(1) through (c)(5).
(2) General procedures.
(i) An arrest shall be made only when the designated immigration officer has reason to believe that the person to be arrested has committed an offense against the United States or is an alien illegally in the United States.
(ii) A warrant of arrest shall be obtained except when the designated immigration officer has reason to believe that the person is likely to escape before a warrant can be obtained.
(iii) At the time of the arrest, the designated immigration officer shall, as soon as it is practical and safe to do so:
(A) Identify himself or herself as an immigration officer who is authorized to execute an arrest; and
(B) State that the person is under arrest and the reason for the arrest.
(iv) With respect to an alien arrested and administratively charged with being in the United States in violation of law, the arresting officer shall adhere to the procedures set forth in 8 CFR 287.3 if the arrest is made without a warrant.
(v) With respect to a person arrested and charged with a criminal violation of the laws of the United States, the arresting officer shall advise the person of the appropriate rights as required by law at the time of the arrest, or as soon thereafter as practicable. It is the duty of the immigration officer to assure that the warnings are given in a language the subject understands, and that the subject acknowledges that the warnings are understood. The fact that a person has been advised of his or her rights shall be documented on appropriate Department forms and made a part of the arrest record.
(vi) Every person arrested and charged with a criminal violation of the laws of the United States shall be brought without unnecessary delay before a United States magistrate judge, a United States district judge or, if necessary, a judicial officer empowered in accordance with 18 U.S.C. 3041 to commit persons charged with such crimes. Accordingly, the immigration officer shall contact an Assistant United States Attorney to arrange for an initial appearance.
(vii) The use of threats, coercion, or physical abuse by the designated immigration officer to induce a suspect to waive his or her rights or to make a statement is prohibited.
... and ...
(f) Site inspections.
(1) Site inspections are Border and Transportation Security Directorate enforcement activities undertaken to locate and identify aliens illegally in the United States, or aliens engaged in unauthorized employment, at locations where there is a reasonable suspicion, based on articulable facts, that such aliens are present.
(2) An immigration officer may not enter into the non-public areas of a business, a residence including the curtilage of such residence, or a farm or other outdoor agricultural operation, except as provided in section 287(a)(3) of the Act, for the purpose of questioning the occupants or employees concerning their right to be or remain in the United States unless the officer has either a warrant or the consent of the owner or other person in control of the site to be inspected. When consent to enter is given, the immigration officer must note on the officer's report that consent was given and, if possible, by whom consent was given. If the immigration officer is denied access to conduct a site inspection, a warrant may be obtained.
(3) Adequate records must be maintained noting the results of every site inspection, including those where no illegal aliens are located.
(4) Nothing in this section prohibits an immigration officer from entering into any area of a business or other activity to which the general public has access or onto open fields that are not farms or other outdoor agricultural operations without a warrant, consent, or any particularized suspicion in order to question any person whom the officer believes to be an alien concerning his or her right to be or remain in the United States.
(g) Guidelines. The criminal law enforcement authorities authorized under this part will be exercised in a manner consistent with all applicable guidelines and policies of the Department of Justice and the Department of Homeland Security
This is just a relatively small portion.