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Don't give in to their sulking. Address the situation if the problem continues. Get some space. Don't take their behaviour to heart.
This can be difficult, but do your best to go about your day normally and pleasantly. Don't spend all your time trying to get them to talk or please them once they've started sulking. Show them that their sulking doesn't elicit the extra attention they want, and hopefully they'll stop sulking or do it less in the future. Instead of giving in, just smile at them, be polite, and stick to your normal routine. Don't enable their behavior. Letting your significant other sulk will only hurt both of you in the relationship. If they keep sulking, address the problem and be upfront without giving them the response they want. Repeatedly asking them what's wrong will only reinforce their behaviour. Make sure to acknowledge their sulking, but don't give in to it. Instead of asking what's wrong, tell them something like "I know you're upset. I'm not sure why, but I'm willing to talk about it whenever you're ready." If they hang around in stony silence, waiting for you to ask them what's wrong or give them attention, remove yourself from the situation. Go to a different room and read for a bit, or go on a walk and get some fresh air. Being around a sulker can bring your mood down as well. Do your best to keep your spirits up and don't let their sulking affect you too much. If the negativity really starts to get to you, try emotionally detaching from the situation a bit and simply observe. While detaching and observing, think to yourself, "It's a shame that they are so unhappy. I hope we can work this out in the future when they're ready." Remind yourself that the sulking is their problem, not yours. Tell yourself that you don't have to be around their behaviour, whether that means leaving for a while or breaking up for good. You don't have to stick with them forever. Don't be afraid to enforce your boundaries in the relationship. Don't let the other person's sulking manipulate or pressure you into giving in.