The major challenge that the Spanish Empire had to face was the challenge from other European nations. Other European countries wanted to gain their own empires and to eat away at Spanish domination. As they attempted to do these things, they presented major challenges to the Spanish Empire.
Two major challenges to the Spanish came from the English and the Dutch. The English were competing with the Spanish for empire in the New World. In addition, English pirates and privateers were preying on the ships that carried the wealth of Spain's empire back to the mother country. The major Dutch challenge to the Spanish Empire came from their attempts to break free from that empire. The Dutch wars of independence took a major toll on the Spanish in terms of money and effort.
These sorts of challenges from European countries inside and outside of the empire were a major threat to the power of the Spanish Empire.