Thanx for the re-post, Tim. You have two different finishes, with different resolutions for each;
In the carpeted area, carefully locate each squeak by repeatedly putting pressure down with one foot, then releasing, moving your foot a little at a time & listening carefully until you feel your right on top of the squeak. First using your hand, start a 2"-2½" galvanized trim nail(the galvanized will help "grab" the wood, and trim nail because it has almost no head), right through the carpet & pad, working the tip a little to help the nail find it's way through the carpet backing. Once it hits the wood subfloor, begin driving the nail with a hammer. If it misses the floor joist you'll feel it slip, and you can pull it back up and move it slightly & try again. Once you've found the joist, drive it down to the carpet surface & finish by driving it on through the carpeting with a nail set until seated. Lay something on this point to visually keep note of it. Locating the next squeak; you can now measure from the first nail, perpendicular to walls & joists, to more closely find the other joists & nail the loose points in the same way. If the floor joists are exposed from below you can also find errant nails & adjust your aim from them.
This causes no damage to the carpeting, just be sure you've seated the nail into the subflooring. It's also a good idea to nail at a slight angle for additional "grab." You can also do this with trumpet-head screws, but you must be very careful doing so, as the screw will try to grab carpet yarn on the way down & can pull it loose.
If the joists are exposed from below, you can also find a squeak & have someone stand on it while you attach a metal "L" bracket to the joist and subfloor from below, but the method of nailing from above is usually the more successful process.
Now to the vinyl-covered area;
Use the same approach to locating the squeaks, but have one person above to activate them, while another is below the floor. If you can visibly see or feel the subfloor moving at the joist, then use a metal "L" bracket, or a bead of construction glue at the joint, then forcing the glue between the joist & flooring while the person above applies & releases pressure (helping to work the glue in between the joist & floor.
If there is no visible motion, it's likely that the squeak is loose underlayment, rather than subfloor, and this becomes tougher to resolve. If you're sure of the subfloor thickness, you can drill upwards through it with a brad point bit to the underside of the underlayment, then inject wood glue using a large hypodermic (no needle). Work the area from above while glueing, as before, then set something heavy on the area while the glue dries. Be VERY careful using this method to not drill through the underlayment; it's often only 1/4" thick. Put a piece of tape on the drill bit as a marker at the appropriate depth to ensure you don't over-drill, or drill through a wood block which only allows the right amount of bit to continue on through the floor. (In example, if the subfloor is 1/2" thick, set your drill bit in the chuck to expose 2" of bit & drill through a 1-1/2" thick board placed against the floor).
Your other option for vinyl is to use the same method as for carpeting, but with small finish nails. This is less desireable, because it does mean putting holes in your finish floor, but done properly the nail points do totally disappear. The vinyl does stretch around the nail, then relax to cover it, but the holes must then be sealed with a vinyl seam sealant. I'd advise against this method unless you felt comfortable with doing such.