The ICAO procedures for "loss of, or significant reduction in, the required navigation capability when operating in an airspace where the navigation performance accuracy is prerequisite to the safe conduct of flight operations" are contained in ICAO Document 4444, Amendment 2 § 15.2.1 and summarized in International Abnormal Procedures / Special Procedures for In-flight Contingencies in Oceanic Airspace.
North Atlantic exceptions to these procedures are contained in ICAO Document 7030 for various regions and in the ICAO NAT Doc 007, North Atlantic Operations and Airspace Manual for MNPS airspace. The North Atlantic exceptions are also provided, below.
There are also other exceptions throughout the world which you can theoretically get from ICAO Doc 7030, but that document is rarely up-to-date. A snapshot of what the most recent version, as of 2015, is included below. You should also make liberal use of your Jeppesen Airway Manual state pages and NOTAMS.
What follows are quotes from the relevant regulatory documents, listed below, as well as my comments in blue.
7.2.1 An aeroplane shall be provided with navigation equipment which will enable it to proceed:
except when, if not so precluded by the appropriate authority, navigation for flights under the visual flight rules is accomplished by visual reference to landmarks.
7.2.2 For operations where a navigation specification for performance-based navigation has been prescribed, an aeroplane shall, in addition to the requirements specified in 7.2.1:
Note.— Information on performance-based navigation, and guidance concerning the implementation and operational approval process, are contained in the Performance-based Navigation (PBN) Manual (Doc 9613). This document also contains a comprehensive list of references to other documents produced by States and international bodies concerning navigation systems.
7.2.3 For flights in defined portions of airspace where, based on Regional Air Navigation Agreement, minimum navigation performance specifications (MNPS) are prescribed, an aeroplane shall be provided with navigation equipment which:
Note.— The prescribed minimum navigation performance specifications and the procedures governing their application are published in the Regional Supplementary Procedures (Doc 7030).
7.2.4 For flights in defined portions of airspace where, based on Regional Air Navigation Agreement, a reduced vertical separation minimum (RVSM) of 300 m (1 000 ft) is applied between FL 290 and FL 410 inclusive, an aeroplane:
7.2.9 The aeroplane shall be sufficiently provided with navigation equipment to ensure that, in the event of the failure of one item of equipment at any stage of the flight, the remaining equipment will enable the aeroplane to navigate in accordance with 7.2.1 and, where applicable, 7.2.2, 7.2.3 and 7.2.4.
Note.— Guidance material relating to aircraft equipment necessary for flight in airspace where RVSM is applied is contained in the Manual on Implementation of a 300 m (1 000 ft) Vertical Separation Minimum Between FL 290 and FL 410 Inclusive (Doc 9574).
[ICAO NAT Doc 007, ¶12.1.2] For unrestricted operation in MNPS Airspace an approved aircraft must be equipped with a minimum of two fully serviceable LRNSs. MNPS approved aircraft which have suffered any equipment failures prior to NAT entry that result in only a single LRNS remaining serviceable may still be flight planned and flown through the MNPS Airspace but only on specified routes established for this purpose.
[ICAO NAT Doc 007, ¶12.1.3] If after take-off, abnormal navigation indications relating to INS or IRS systems occur, they should be analysed to discover their cause. Unless the flight can proceed safely using alternative approved navigation sources only, the pilot should consider landing at the nearest appropriate airfield to allow the problem to be fully investigated, using technical assistance if necessary. Under no circumstances should a flight continue into oceanic (MNPS) Airspace with unresolved navigation system errors, or with errors which have been established to have been caused by inertial platform misalignment or initial position insertion error.
[ICAO NAT Doc 007, ¶12.1.6] With only two systems on board, identifying the defective unit can be difficult. If such a situation does arise in oceanic airspace any or all of the following actions should be considered:
[ICAO NAT Doc 007, ¶12.1.7] Occasions may still arise when distance or cross track differences develop between systems, but the crew cannot determine which system is at fault. The majority of operators feel that the procedure most likely to limit gross tracking errors under such circumstances is to fly the aircraft half way between the cross track differences as long as the uncertainty exists. In such instances, ATC should be advised that the flight is experiencing navigation difficulties so that appropriate separation can be effected if necessary.
[ICAO NAT Doc 007, ¶12.1.8] Operations or navigation manuals should include guidelines on how to decide when a navigation system should be considered to have failed, e.g. failures may be indicated by a red warning light, or by self diagnosis indications, or by an error over a known position exceeding the value agreed between an operator and its certifying authority. As a generalisation, if there is a difference greater than 15 NM between two aircraft navigation systems (or between the three systems if it is not possible to detect which are the most reliable) it is advisable to split the difference between the readings when determining the aircraft's position. However, if the disparity exceeds 25 NM one or more of the navigation systems should be regarded as having failed, in which case ATC should be notified.
[ICAO NAT Doc 007, ¶12.2.1] Some aircraft carry triplex equipment (3 LRNSs) and hence if one system fails, even before take-off, the two basic requirements for MNPS Airspace operations may still be met and the flight can proceed normally. The following guidance is offered for aircraft having state approval for unrestricted operations in MNPS airspace and which are equipped with only two operational LRNSs.
The reason North Atlantic procedures are different is that the North Atlantic can become incredibly crowded and they would rather you stay on the tracks if you can safely (and accurately) do so. The procedures that follow assume you only started with two LRNS. If you had three and lose one, you are good to go. If you had three and lose two, the following also applies to you.
[ICAO NAT Doc 007, ¶12.2.2] The pilot must consider:
These Blue Spruce Routes are listed in ICAO NAT Doc 007 and your Jeppesen Airway Manual Atlantic Planning Charts.
[ICAO NAT Doc 007, ¶12.2.4] The pilot must consider:
[ICAO NAT Doc 007, ¶12.2.5] Once the aircraft has entered oceanic airspace, the pilot should normally continue to operate the aircraft in accordance with the Oceanic Clearance already received, appreciating that the reliability of the total navigation system has been significantly reduced.
[ICAO NAT Doc 007, ¶12.2.8 The pilot should:
Note: This procedure also applies when a single remaining system gives an indication of degradation of performance, or neither system fails completely but the system indications diverge widely and the defective system cannot be determined.
[ICAO NAT Doc 007, ¶12.2.9] A characteristic of the navigation computer system is that the computer element might fail, and thus deprive the aircraft of steering guidance and the indication of position relative to cleared track, but the basic outputs of the IRS (LAT/LONG, Drift and Groundspeed) are left unimpaired. A typical drill to minimise the effects of a total navigation computer system failure is suggested below. It requires comprehensive use of the plotting chart.
Theoretically, you should be able to examine ICAO Doc 7030 to come up with regional differences and call it good. Unfortunately this document is rarely up to date and you should closely study the state pages in your Jeppesen Airway Manual as well as any NOTAMS. If you do examine "Chapter 9. Special Procedures" for each region in ICAO Doc 7030 as of 2015, here is a summary of what you will find:
Portions of this page can be found in the book International Flight Operations, Part VI, Chapter 3.
ICAO Annex 6 - Operation of Aircraft - Part 1 Commercial Aircraft, International Standards and Recommended Practices, Annex 6 to the Convention on International Civil Aviation, Part I, July 2010
ICAO Annex 6 - Operation of Aircraft - Part 2 General Aviation, International Standards and Recommended Practices, Annex 6 to the Convention on International Civil Aviation, Part II, July 2008
ICAO Doc 7030 - Regional Supplementary Procedures, International Civil Aviation Organization, 2008
ICAO NAT Doc 007, North Atlantic Operations and Airspace Manual, Edition 2014/2015