Unless you are getting a code for the evap system, your gas cap is fine.

Do the code come up immediately after you reset the code or does it take some time?

When you cold start the engine do you feel the misfire, or do the misfire come on after the SES light?

Do you have a scanner that can view real time data such as short term and long term fuel trim?

Even though you cleaned the MAF, you could have a Bad MAF and no amount of cleaning will get it working right. The app Torque can provide live data of the MAF. Don't bother buying a cheap MAF from eBay or Amazon. Go to a junk yard or buy a used Siemens MAF.

No evap codes
After clearing the codes, the failures returning are very random. Some times as shortly as two miles and as long as 40 miles.
No noticeable rough starts.
I get the SES light at the same time it noticeably runs rough. Ill pull over and turn off the car, wait a few seconds, restart and all is good until next failure.

I do not have a scanner. Only a OBDII reader. Any suggestion on a scanner/diagnostic tool?

Thanks for the reply and suggestions.

You don't have a normal misfire in the sense that the cylinder is not firing, but you have a misfire in the sense that the computer is able to detect that the cylinder is not producing enough power compare with the other cylinders so it will count the number of time the "misfires" occurs and once the number exceeds the limit, it will shut down the injector to that cylinder and throw up the SES light. At that point since the cylinder is no longer receiving gas the misfire is noticeable.

Without being able to perform further testing on your engine on my part, my best guess will be that the MAF is probably bad. See my previous comment about buying a MAF.

Waiting for someone to chime in about doing the unplug the MAF test which I don't know if it really works, which is why I didn't ask you to perform that test.

Still fighting the good fight with no gains.
Still getting the random misfire codes with fuel cut-off.

Last night I removed and cleaned the injectors. I cleaned them with carb cleaner then cleaned them in a ultra sonic cleaner. Rigged up an electrical connector, switch and battery, opened each injector electrically and blow pressurized carb cleaner through them. Reinstalled.

No change. Still getting the codes.

I have ordered a new MAF. Maybe thats it?????

My list of things that I have repaired or replaced and NOT fixed the problem is getting longer. Must mean that I am somewhat closer to the solution.

Again, I appreciate any help or advise offered.

PB

Hopefully you ordered a Siemens MAF. The cheap one on eBay or Amazon is hit or miss with more misses then hits.

Are you using the stock air filter system or have you installed a CAI system?

Upallnight, I took your advise and ordered the Siemens/VDO MAF, and I have the stock air filter.
PB

Well the new MAF sensor did not fix it.
Drove it today into town and got a P1347 code and SES light. (misfire #3 with fuel Cut-Off)

Is it possible that the electronically controlled thermostat could be causing this problem?
These misfires started started soon after I replaced the thermostat.

Thanks for any help.

It appears your misfire has gone from cylinder 5 to cylinder 3. Has the misfire always been sporadic? I thought from your previous postings that you had a misfire in cylinder 5. A cam sensor could be the other cause for the sporadic misfires.

P1347 BMW Misfire Cylinder 3 With Fuel Cut-off OBDII Engine Light Trouble Code | Engine-Codes.com

The cam sensor wiring/connector could had been knocked loose while you were replacing the thermostat.

Here's a video on replacing the cam sensor.

https://www.youtube.com/watch?v=ImJsKtMcgPE

I would also check the connector to the VANOS solenoid.

Thanks, Ill look into it.