Powers of The High Court
The Constitution directs the government that High Court shall have power, throughout in relation to it jurisdiction, to issue to any person or authority, including in appropriate cases, any Government, directions, orders or writs, for the enforcement of any of the rights conferred by Part III and for any other purpose also.

Definition
Article 226. Power of High Courts to issue certain writs:
(1) Notwithstanding anything in article 32, every High Court shall have power, throughout the territories in relation to which it exercised jurisdiction, to issue to any person or authority, including in appropriate cases, any Government, within those territories directions, orders or writs, including 3[writs in the nature of habeas corpus, mandamus, prohibition, quo warranto and certiorari, or any of them, for the enforcement of any of the rights conferred by Part III and for any other purpose].

(2) The power conferred by clause (1) to issue directions, orders or writs to any Government, authority or person may also be exercised by any High Court exercising jurisdiction in relation to the territories within which the cause of action, wholly or in part, arises for the exercise of such power, notwithstanding that the seat of such Government or authority or the residence of such person is not within those territories].

(3) Where any party against whom an interim order, whether by way of injunction or stay or in any other manner, is made, or in any proceedings relating to, a petition under clause (l), without-
(a) furnishing to such party copies of such petition and all documents in support of the plea for such interim order; and
(b) giving such party and opportunity of being heard, makes and application to the High Court for the vacation of such order and furnishes a copy of such application to the party in whose favor such order has been made or the counsel of such party, the High Court shall dispose of the application within a period of two Weeks from the date on which it is received or from the date on which the copy of such application is so furnished, whichever is later, or where the High court is closed on the last day of that period, before the expiry of the next day afterwards on which the High Court is open; and if the application is not so disposed of, the interim order shall, on the expiry of that period, or, as the case may be, the expiry of the said next day, stand vacated.
(4) The power conferred on a High Court by this article shall not be in derogation of the power conferred on the Supreme Court by clause(2) of article 32.

Note- Powers of the High Court to issue writs under article 226 are wider than that of the Supreme Court. It is not confined to fundamental rights, but extends to all cases where the breach of a right is alleged. The writ may be issued for enforcement of fundamental rights or for ‘any other purpose. AIR 1952 SC 12.

Certiorari and prohibition- The ground for issuing certiorari and prohibition are as under:
(a) order without jurisdiction attracts certiorari.
(b) an error of law apparent on the face of the record.
(c) irregularity in procedure.
(d) erroneous decision;

Interfere in investigation-
While the court should not nomially interfere with criminal investigation, it may prohibit the continuance of a criminal proceeding at any stage before its conclusion, in extraordinary cases, e. g., where the proceeding was launched by a person who was not competent to make the F .I.R. or to institute prosecution under the relevant law; State of Harjyana v. Bhajan, AIR 1992 SC 604

Writ to Bail-
Although the High Court has jurisdiction to entertain prayer for bail in writ jurisdiction, yet, such jurisdiction should be exercised sparingly having regard to judicial discipline and comity of courts; Kartar Singh v. State of Punjab, (1994) 3 SCC in the Bimla v. State of Bihar, (1994) 2 SCC 8 it is said by the Supreme Court that- after rejection of bail by the Magistrate, High Court grants bail, Supreme Court may interfere.

The Constitution of India mentions the writ of habeas corpus in articles 32 and 226.
(a) Habeas corpus
The exceptional cases in which threatened detention may give rise to the writ, are where the court is satisfied about one of the following circumstances—
(a) illegal order of detention
(b) arrest or detention order made against a wrong person; or
(c) arrest or detention order made for a wrong purpose; or
(d) arrest or detention order is passed on baseless grounds; or
(e) arrest or detention order made without authority, in law

Prohibition- The Supreme Court and the High Courts have power to issue writs, including a writ of prohibition. A writ of prohibition is normally issued only when the inferior Court or Tribunal—
(a) proceeding without or in excess of jurisdiction,
(b) order in violation of rules of natural justice,
(c) proceeding is ultra vires the constitution, or
(d) against fundamental rights.

Difference in powers of the High Court and the Supreme Court:
Supreme Court is the apex court of India has unlimited powers in the field of judiciary. Even then, amongst the courts below, the High Court has given very wide powers to order for enforcement of (i) fundamental right including (ii) non-fundamental (m) ordinary legal rights, except to decide question of law relating to interpretation of the Constitution etc.