The phrase "evil spirit(s)" occurs 14 times (in 13 verses) in the Bible, in 8 of which the evil spirits were sent directly and explicitly by God.
The first evil spirit was sent by God to kill people in war.
Then God sent an evil spirit between Abimelech and the men of Shechem; and the men of Shechem dealt treacherously with Abimelech. Judges 9:23
God sent the next evil spirit to cause Saul to have a mental breakdown. (But luckily, David plays his harp and makes Saul's "evil spirit from the Lord" go away.)
But the Spirit of the LORD departed from Saul, and an evil spirit from the LORD troubled him. And Saul's servants said unto him, Behold now, an evil spirit from God troubleth thee. Let our lord now command thy servants, which are before thee, to seek out a man, who is a cunning player on an harp: and it shall come to pass, when the evil spirit from God is upon thee, that he shall play with his hand, and thou shalt be well. 1 Samuel 16:14-16
And it came to pass, when the evil spirit from God was upon Saul, that David took an harp, and played with his hand: so Saul was refreshed, and was well, and the evil spirit departed from him. 1 Samuel 16:23
Then God sent an evil spirit to cause Saul to prophesy and sit with a javelin in his hand.
And it came to pass on the morrow, that the evil spirit from God came upon Saul, and he prophesied in the midst of the house: and David played with his hand, as at other times: and there was a javelin in Saul's hand. 1 Samuel 18:10The other 6 "evil spirit" occurrences are in the New Testament. The Bible doesn't say where the evil spirits came from, but they are usually associated with some type of illness (Which come from God, right?).
And the evil spirit from the LORD was upon Saul, as he sat in his house with his javelin in his hand: and David played with his hand. 1 Samuel 19:9
And in that same hour he cured many of their infirmities and plagues, and of evil spirits; and unto many that were blind he gave sight. Luke 7:21
And certain women, which had been healed of evil spirits and infirmities, Mary called Magdalene, out of whom went seven devils, Luke 8:2
So that from his body were brought unto the sick handkerchiefs or aprons, and the diseases departed from them, and the evil spirits went out of them. Then certain of the vagabond Jews, exorcists, took upon them to call over them which had evil spirits the name of the Lord Jesus, saying, We adjure you by Jesus whom Paul preacheth. ... And the evil spirit answered and said, Jesus I know, and Paul I know; but who are ye? And the man in whom the evil spirit was leaped on them, and overcame them, and prevailed against them, so that they fled out of that house naked and wounded. Acts 19:12-16
So the Bible is fairly clear on evil spirits: they are either sent directly by God or their origin is unknown. The Bible never attributes evil spirits to Satan.
(Thanks to Sternwallow at the Raving Atheists Forum for this one.)