The various Administrative Tribunals in Manitoba have information on their websites about process:

Residential Tenancies Branch
Policies and Procedures Handbook

The Court of Appeal

The Court of Appeal has information about appealing a Residential Tenancies Commission case to the Court of Appeal

You may also want to view Manitoba Courts video:

Seeking Leave to Appeal

http://www.manitobacourts.mb.ca/video-library/

Social Services Appeal Board 
Policy and Procedure Manual

Workers Compensation Board
Information about Appealing a Decision