I have a 1999 Saturn SL1 four door. I am having problems starting it sometimes.

So when it doesnt start is the starter turning the engine over?

Could be starter is going, brushes are worn out. When it wont spin, tap with hammer/tool if that makes it work it's shot.

I would have the battery test along with the alternator also make sure the battery cables ar clean and tight. Just to rule those out. Then when it wont start see if there power at the purple wire at the starter with the key in the start position. If so bad starter usually the solenoids fail on those. Its pretty common for q bad starter to be a intermintant no crank. Let me know what you find.

Before starting remove all jewelery like rings watches etc. Then first step remove the battery cables take off the black negative cable first its a 8mm. Becarefull not to touch the wrench to the positive post. Then remove the positive cable inspect both cables for corrossion. If ther is corrossion clean with a side post battery brush and baking soda and water. You can also soak the battery cable enda in baking soda and water. Next put the positive battery cable back on the battery. Then put the negative battery cable back on. Becarefull again to not touch the positive battery post. Then next step go to the auto parts store have them test the battery and alternator. While your at the auto parts store if you dont have a test light or multimeter get one. Best bet would be a test light let me know when your done with the auto parts store and I will give you the rest of the steps.

Saturn starter wont operate at all if brushes are shot. No noise at all. If you tap it and it works, it's the brushes. Bad connections can account for problem also.

Thats strange I have seen the bedixs kick out without the starter spining on those saturn starters.I would have thought that would be bad brushes or a bad armature. Doesny make sense to me that the power goes thru the starters brushes then thru the solenoid. Especailly after seeing the bendix kick out with the starter not spinning. Thats just my expernice with those starters I have seen a lot of bad ones over the those.