“签署” is primarily used in formal, written contexts or in serious, formal speech. It is the standard term in business, law, and diplomacy.
Business Context: Used for signing contracts (合同, hétong), agreements (协议, xiéyì), and letters of intent (意向书, yìxiàngshū). It signifies that a deal is officially made.
Legal & Governmental Context: This is the word used when officials sign laws (法案, fǎ'àn), treaties (条约, tiáoyuē), joint statements (联合声明, liánhé shēngmíng), or other official decrees.
Formality: Highly formal. You would almost never use it in casual, everyday conversation about mundane tasks. For signing a receipt or a guestbook, you would use “签名” (qiānmíng).