The verb already shows who is speaking, so аз (I) is optional, in writing as much as in speech. Leaving it out is the normal, neutral way to say it.

• "Щастлив съм." (I am happy.) Neutral, the everyday form.
• "Аз съм щастлив." (I am happy.) Stresses I, for example in contrast to someone else: "Ти си тъжен, а аз съм щастлив." (You are sad, but I am happy.)

съм can never start a sentence, because it is a clitic (a short word that leans on the word before it). Without аз, it moves after the next word:

• "Аз съм студент." → "Студент съм." (I am a student.) Never "Съм студент."
• "Аз съм уморен." → "Уморен съм." (I am tired.)