Here is a breakdown of what inspections are required for each certificate level and each added rating for operations under FAR 91. While there are going to be some minor nuances to this list, this is basically it for 99% of the piloting you will do.
Private Pilot ASEL, AMEL
There are five (5) inspection requirements related to Private Pilot operations:
Annual Inspection - must have been performed within the last 12 calendar months. (reference FAR 91.409 (a)) This must be performed by an A&P Mechanic with a Inspection Authorization (called an IA) or it can be performed by a FAA Certified Repair Station (called a CRS).
ELT Battery - This battery must not be expired - check the expiration date in the aircraft logbooks. (reference FAR 91.207 (c)). Only a certified aircraft mechanic or a CRS can replace this battery. When looking for this record, start looking for it with the latest entries in the Aircraft Logbook. Look back in time from there.
ELT - This ELT must have been inspected within the last 12 calendar months. (reference FAR 91.207 (d)). Only a certified aircraft mechanic a CRS can perform this inspection.
Transponder - This device must have been inspected within the last 24 calendar months. (reference FAR 91.413) Only a CRS can perform this inspection. This inspection is only required if the aircraft has a transponder installed.
Airworthiness Directives (AD's) - All AD's must be in current compliance. (reference FAR 91.403 (a)). There must be a list of all AD's that apply to an aircraft be kept with the aircraft records. Only the persons authorized as mentioned in the AD are allowed to perform this inspection. The authorized personnel are typically the CRS or a A&P mechanic.
Notice that there is no requirement for a 100 hour inspection when a Student or Private Pilot is operating an aircraft unless the owner of the aircraft is providing flight instruction to the Student or Private Pilot.
Scenario #1: You are renting the school's airplane and are flying it without the school's instructor onboard (solo). There is no 100 hour inspection required.
Scenario #2: You are renting the school's airplane and are flying it with the school's instructor (A.K.A. their employee) onboard (it's a dual flight). A 100 hour inspection is required.
Scenario #3: You are taking the FAA practical test for a pilot certificate and/or rating. The FAA Designated Pilot Examiner is not acting as an instructor during the flight. There is no 100 hour inspection required.
Instrument Airplane
In addition to the above listed Private Pilot requirements, there are two (2) additional required inspections when operating on an IFR flight plan:
VOR inspection - If used during an IFR flight, the VOR must have been previously inspected within the last 30 days. (reference FAR 91.171) Pilots are allowed to perform this test.
Single VOR Receiver ground test - +/-4° error, Airborne test +/-6° error allowed.
Select a published airborne check point or tune in a published VOT signal or;
Select a VOR radial that lies along the centerline of an established VOR airway and select a prominent ground point along the selected radial (preferably more than 20 nautical miles from the VOR) and maneuver the aircraft directly over the point at a reasonably low altitude.
Dual VOR Receiver ground test +/-4° error, Airborne test +/-6°.
In lieu of the above. the Dual VOR test can be easily accomplished by cross checking both VOR receivers in the aircraft and they must be within +/-4° of each other. This test can be accomplished anywhere a VOR (or VOT) signal is being received, it does not need to be performed along an airway.
This inspection (test) must be entered into a record that contains the Date, Place, Bearing Error and the Signature of the person performing the test.
Contrary to popular belief, this record is not required to be kept in the aircraft.
Static system and Mode C Altitude Encoder Inspection - These devices must have been inspected within the last 24 calendar months. (reference FAR 91.411). Only a CRS can perform this inspection.
Commercial Pilot ASEL, AMEL
In addition to the above listed Private Pilot requirements, there is one (1) additional required inspection:
100 Hour Inspection - must have been performed within the previous 100 hours of operation since the last Annual inspection or the last 100 hour inspection. (reference FAR 91.409 (b)). This must be performed by an A&P Mechanic or it can be performed by a FAA Certified Repair Station (CRS).
When operation for hire or when the owner of the aircraft is providing flight instruction in the aircraft, it must have this inspection. This means when paying passengers and/or cargo are onboard it must be in compliance with this rule. Also, when the owner (such as a flight school) in giving flight instruction in the aircraft (a dual flight) in must comply with this rule. Solo and rental flights are exempt from this rule.
Exceeding the 100 hour time limit - This
grace period of up to 10 hours is allowed for the limited purpose of
traveling to a place where the 100 hour inspection can be performed.
The aircraft is still considered airworthy during this time, the
paying passengers and/or cargo are allowed to be onboard during this
time period where 100 hours have been exceeded. The amount to time
the aircraft exceeded the 100 hour time limit is simply subtracted
from the next 100 hour period.
Example - 105 hours flown, next
100 hour will be due in 95 hours.
If instead of a 100 hour inspection being performed and only 95 hours being allowed to the next 100 hour inspection, an alternative is to have an Annual inspection performed instead of a 100 hour inspection. The Annual inspection "resets" the rules to allow a full 100 hours of operation until the next 100 hour inspection.
In the event the aircraft has accumulated more than 110 hours since the last Annual or 100 hour inspection, it cannot be used for hire or for flight instruction by the owner until it receives an Annual inspection.
There is no difference between how an Annual and a 100 hour inspection are performed (reference FAR 43 appendix D), the only difference is who authorized to perform the inspection (IA or CRS)