Kenmore 665 68610891- built-in 1998 model

This is normal. Kenmore microwaves will stay running with fan and light on for a short time after food is cooked. The reason is to air out the microwave after cooking. Things like pop corn, Bacon etc. have strong smells after cooking. The ventilation helps to remove odors.

Actually, that's a feature! If you cook much on the range (stove) beneath the microwave, the heat will cause the microwave bottom to heat up quite a bit (just touch it!), and the microwave will turn on it's exhaust fan in an attempt to save itself from burning up. It will shut off in about 15 minutes after you turn off the source of heat.
But if the fan comes on by itself even when the microwave is cool (or at least not hot), then you probably have a defective fan thermostat. This thermostat is located behind the control console on the inside bottom of the microwave.

If you need further help, reach me via phone at https://www.6ya.com/expert/ronald_e6b9572cebe3ad27

Actually, that's a feature! If you cook much on the range (stove) beneath the microwave, the heat will cause the microwave bottom to heat up quite a bit (just touch it!), and the microwave will turn on it's exhaust fan in an attempt to save itself from burning up. It will shut off in about 15 minutes after you turn off the source of heat.
But if the fan comes on by itself even when the microwave is cool (or at least not hot), then you probably have a defective fan thermostat. This thermostat is located behind the control console on the inside bottom of the microwave.

If you need further help, reach me via phone at https://www.6ya.com/expert/ronald_e6b9572cebe3ad27

With the full model number, chances are we could have helped you more.
If the cooktop lights recently failed, it may need circuit board repair.

Or it may be a failed hood TCO.

You can find links to helpful exploded view diagrams and part ordering help here.

I recommend that you bookmark that link to your favorites.

sounds like the cooling fan has came adrift or stopped working , this will cause the magnatron ( the microwave generator ) to overheat tripping the over temperature stat ,so fix the cooling , but be sooooooooooooo carefull in there , microwaves + really high voltages , not really a job for a non electrician/microwave tech

Can you hear the cooling fan running if not its getting hot a cutting off. once cooled it starts back up

I have the exact same problem with my Kenmore Microwave model 721 that we bought in October of 2005. The Fan will just randomly turn itself on even when we are not cooking. It has been doing this for the last few months and is starting to become more frequent. There is no way to turn it off except to unplug it for several hours.
Was there a recall for this defect ?

Your hood fan thermostat is probably stuck "on" and will have to be replaced.

There should be a "mini-manual" (tech sheet) hidden inside the unit behind the control panel or hidden on the left side behind the grille, which is very helpful when troubleshooting, testing, and locating components.

At our Web site, we have a video available showing how to remove a typical over the range control panel assembly in under 5 minutes.

You can usually find helpful exploded view diagrams and order parts by entering your full model number here.

We're happy to help you with free advice and we'd appreciate your thoughtful rating of our answer.

I HAVE KENMORE 665/61629 LIGHT STAYS ON. AND DO NOT COOKS. AND THE FAN DOES NOT WORK.

So is it cooking okay?

I would guess that your hood thermostat is failing.

The hood thermostat's job is to turn on the exhaust / vent / hood fan whenever the temperature above the cooktop gets too high.

A thermostat can only trip so many times before it starts to fail. Then it may come on by itself from time to time then finally stay on permanently.

If you or someone you know wants to look into this, we have critical safety, disassembly, and troubleshooting info at our site, and our link is at our listing here on FixYa.

We even have a video on our site that shows you how to safely & easily remove the control panel assembly in less than 5 minutes so you can check out your oven.

You can usually find helpful exploded view diagrams and order parts by entering your full model number here.

There should also be a "mini-manual" hidden inside the unit behind the control panel or hidden on the left side behind the grille, which is very helpful when troubleshooting & testing.

We're happy to help you and we'd appreciate your thoughtful rating of our answer.