Which one is more common and natural? (in spoken and written English)
- Someone has left their bag on the table?
- Someone has left his or her bag on the table?
- Someone has left his bag on the table?
All three sentences are acceptable in both written and spoken English, but each has slightly different meanings.
Using "their" indicates you don't know either the gender or number of people who left the bag on the table.
Using "his or her" indicates you don't know what gender the person who left the bag was, but you know (or assume) it was only one person.
Using "his" indicates you assume the (singular) person who left the bag is male.
If I were to guess the most common spoken usage for this I would personally use the sentence "Someone left their bag on the table."