Can my training in a Sport aircraft with a Sport instructor be used to fulfill the requirements to become a Private Pilot?
As an example, assume I've started my training to be a Sport Pilot, but decided I'd rather be a Private Pilot.
Absolutely, you can log just about any flight time in an aircraft!
The definition of flight time is in 14 CFR 1.1 and says:
Flight time means:
(1) Pilot time that commences when an aircraft moves under its own power for the purpose of flight and ends when the aircraft comes to rest after landing; or
(2) For a glider without self-launch capability, pilot time that commences when the glider is towed for the purpose of flight and ends when the glider comes to rest after landing.
However, the instruction logged towards a sport pilot rating by a flight instructor certificate with a sport pilot rating (61.413) may not be used toward the requirements of the private pilot certificate.
If the instructor is a CFI certified under 61.181 then the instruction may also be used towards your private pilot certificate but you will need to be proficient to the higher standards, which may require additional training.
For those who already have their Sport Pilot Certificate, take a look at this question: What does a sport pilot have to do to get a private pilot certificate?