I'd like to add 1/8" to my subfloor so that my 3/4" hardwood is level with my tile. My current subfloor is 3/4" OSB...any suggestions on what to use and how to fasten it?

If you are performing a nail down application you can layer a couple of strips of vinyl flooring scraps. I use one approximately 3 " wide and a second approximately 9" wide. The two layers should get you to your 1/8" needed to meet your tile.

If you are performing a glue down application you will need to ramp your floor using a leveling compound or you could use a flex tile thinset.

I was thinking more along the lines of adding 1/8" to the entire floor.....I gather that by the strips of linoleum, you were ramping the hardwood up to the level of the tile. I'd rather keep the hardwood as level as possible.

The hardwood flooring store recommended using some 1/8" plywood over the entire floor. Should I be glueing and stapling since it is so thin?

You would never notice the ramping of the floor at the transitions.

If you lay a straight edge on your floor now. You will find many depressions that are 1/8" or greater. I would be amazed if your floor did not have many depressions approaching 1/4"

It is the exception that a floor will not have that kind of variance. The standard for a floating floor is 3/16" of an inch over 10'. Using a 6' straight edge I have never found a single floor that can meet that standard without some sort of sanding and filling.

If you are concerned about the ramp effect. create your strips (assuming you are using a nail down application) wider than I have originally suggested.

I hope this helps.