Who can carry out airworthiness inspections?
I can speak to the FAA requirements.  The basic requirement for the annual inspection (commercial aircraft have additional requirements) comes from 14 CFR Part 91.409  


  §91.409   Inspections.  
  
  (a) Except as provided in paragraph (c) of this section, no person may
  operate an aircraft unless, within the preceding 12 calendar months,
  it has had—
  
  (1) An annual inspection in accordance with part 43 of this chapter
  and has been approved for return to service by a person authorized by
  §43.7 of this chapter; or...


And the related requirement regarding who can perform this is covered in Part 43 and is in two parts; who can perform the work and who can approve the return to service.  

The work must be performed by someone specified in 14 CFR Part 43.3 in which the general case is:


  §43.3   Persons authorized to perform maintenance, preventive
  maintenance, rebuilding, and alterations.   
  
  (a) Except as provided in this section and §43.17, no person may
  maintain, rebuild, alter, or perform preventive maintenance on an
  aircraft, airframe, aircraft engine, propeller, appliance, or
  component part to which this part applies. Those items, the
  performance of which is a major alteration, a major repair, or
  preventive maintenance, are listed in appendix A.
  
  (b) The holder of a mechanic certificate may perform maintenance,
  preventive maintenance, and alterations as provided in Part 65 of this
  chapter.
  
  (c) The holder of a repairman certificate may perform maintenance,
  preventive maintenance, and alterations as provided in part 65 of this
  chapter.


And the approval for return to service must be per 14 CFR Part 43.7  


  §43.7   Persons authorized to approve aircraft, airframes, aircraft
  engines, propellers, appliances, or component parts for return to
  service after maintenance, preventive maintenance, rebuilding, or
  alteration.  
  
  (a) Except as provided in this section and §43.17, no person, other
  than the Administrator, may approve an aircraft, airframe, aircraft
  engine, propeller, appliance, or component part for return to service
  after it has undergone maintenance, preventive maintenance,
  rebuilding, or alteration.
  
  (b) The holder of a mechanic certificate or an inspection
  authorization may approve an aircraft, airframe, aircraft engine,
  propeller, appliance, or component part for return to service as
  provided in Part 65 of this chapter.  


Requirements for mechanics and inspection authorizations are spelled out in 14 CFR Part 65, Subpart D.   

The regulations also allow manufacturers and repair stations among a limited number of others to perform the work as well.  (see links)