As most people are aware, typically LED smart bulbs will come on at full power after a power outage. This is done so that the bulbs will also work if someone just manually throws the switch. Otherwise, if the bulb had been turned off with the app, it might not turn on again if someone flipped the switch a couple of times.
Outlets are different, because you aren’t likely to run into the same confusion, so different manufacturers have handled the situation differently. If this feature will be important to you, make sure you read the model specifications carefully.
In general, GE, Linear/GoControl and Schlage outlets will be in the “off” state after power is restored.
Leviton and Neo Coolcam pocket sockets will all attempt to restore to the previous state from the time the power was cut.
The Zooz Zen15 heavy duty pocket socket is unusual in both form and function. It has a parameter which lets you set it to either always come on after a power outage or always stay off after a power outage. ( it doesn’t remember previous state, you are just picking the start up activity.) It is specifically designed for appliances with motors like freezers and pumps, as long as they are 15 A or lower.
@scodes has tested the following as well:
The Aeotec Smart Switch 5 version 6 restores “previous state”
The Smartthings appliance module restores “previous state”
Please feel free to post other brands if you have confirmed their power outage behavior.