What are some "FAA approved" simulators for home that I can log time on?
Read this answer from AOPA, especially the last 2 paragraphs.
https://www.aopa.org/news-and-media/all-news/2004/november/flight-training-magazine/since-you-asked

Logging sim time
Dear Rod:
I'm a private pilot just a couple of days from taking my instrument checkride. My CFI and I have had discussions about simulator hours. He logs them but does not include them in total time. I've just closed the Federal Aviation Regulations/Aeronautical Information Manual for the umpteenth time and after reading about maximum of 20 here and a credit of 100 there, I'm confused. Maybe you can explain. What is the point of taking advantage of a sim if the hours logged don't count toward my TT?
Thank you,
Mark
Greetings Mark:
Simulator hours can count as training time, but are not considered &quot;flight time&quot; and can't be logged in the flight time (total time) column of your logbook. They can, however, be logged in the &quot;Flight Simulator&quot; column of your logbook. If your logbook doesn't have such a column, then make one using the letters &quot;FS/FTD&quot; which stand for flight simulator/flight training device.
Regarding simulator time counting toward your total time, consider that the commercial certificate allows 50 hours of simulator time to be counted toward the total aeronautical experience requirement of that rating -- see FAR 61.129(i)(1).
Thus, you could meet the commercial requirements of 250 hours of aeronautical experience with 200 hours of flight time and 50 hours of simulator time.
Even if simulator time didn't count toward the total aeronautical experience requirement of a rating, it might still count as training time. For instance, 2.5 hours of appropriate simulator time can be credited toward the total training time required for the private pilot certificate (FAR 61.109 i).
Now, read the federal aviation regulations carefully before running off and trying to log cross-country time in a simulator. The regs specifically state you must meet certain &quot;flight time&quot; requirements in an airplane. You're not getting away with packing a lunch and sitting in front of a simulator for six hours in hopes of logging this as a cross-country flight to meet the private pilot cross-country requirements.
(Yes, I heard of a case where someone actually tried to do this. Perhaps the ruse unraveled when the FAA examiner asked, &quot;Hey, what kind of airplane is a Combat Flight Sim?&quot;)
Also keep in mind that you can only log simulator time when there is an appropriately rated instructor providing dual instruction. In other words, solo time in a simulator can't count toward your private or commercial certificate or instrument rating.
And it goes without saying that a simulator must be approved by the FAA for it to be used as a FS or FTD. (Sorry, but Combat Flight Simulator is only approved for shooting down other simulators, not flight training.)
Rod Machado is a flight instructor, author, educator, and speaker. A pilot for 34 years and a CFI for 30, he has flown more than 8,000 hours and owns a Beech A36 Bonanza. Visit his Web site.

FAA regs on simulators
https://www.faa.gov/about/initiatives/nsp/ac/
List of FAA approved simulators, and the aircraft type simulated
https://www.faa.gov/about/office_org/headquarters_offices/avs/offices/afx/afs/afs800/afs810/media/FAA_Approved_Airplane_ATDs.pdf