... Then you can fix your problem. If you DON'T, you will 'ruin' the look of the stairs by screwing the treads to the riser(s) and stringers from above, even though you're covering them up with carpet. I would NOT do anything unless you can gain access from underneath! You may not heed this advice but it's your house. I'll tell you now, that a home with nice oak stairs, finished or unfinished, covered with carpet or not, adds value. If you 'deface' them (IMO), you lessen the value because it won't look good when they're refinished. (I'd take off the carpet and refinish the oak stairs BEFORE I go and sell the house. And if someone with an 'eye' sees your nails/screws all over the place, you may lose a sale. That's my opinion.)
If you have access from underneath, and you see any wedges, remove them and add some glue and set them back in. If there are any places where you can ADD a wedge, add some glue and tap them in.
If you find a squeeker, glue and screw some blocks in place underneath where the riser meets the tread. Glue and screw the block to the riser first, just a little lower than the upper edge of the riser so that the tread will be pulled down to it by the screw.
You can also install metal "L" shelf-brackets too. Again, mount them a little lower than the upper edge on the riser so that the tread will be pulled down.
FWIW, I have squeeks all over my steps. But you know what, I think of them as a form of 'house alarm'. In the quiet of the night, I'd be able to hear just about anything walking up the steps! Especially someone who doesn't know where the squeeks are. Sure, I may still get 'iced' but at least I'd have dialed 911 before I hit the floor. My best to ya and hope this helps.