Closed or open cell spray foam is the surest most effective insulation on the ceiling. But it should be followed by a 15-minute thermal barrier. Half-inch drywall is the typical barrier. But there are a number of others that may be approved, such as intumescent paint. And some spray foams, in fact, may be approved without a barrier. Be sure to check with your local code enforcement office to be sure what method they approve.

rigid foam insulation in 4 x 8 sheets and cut it to fit the insides of your doors

People do this all the time. However, it must be covered with an approved thermal barrier.

While enforcement of barriers may be spotty in some jurisdictions, you can be sure if there is a fire your insurer will be up to speed.

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In one home where I had the basement totally insulated in closed cell spray polyurethane foam a plumber managed to set just a tiny bit the size of a finger on fire. Everyone working fled immediately from the deadly odour. Thermal barriers are not something to skimp on just because a Code authority may miss it.