Flooring – Should I remove old linoleum when replacing a linoleum floor

We are assuming here that the noise is caused by hard shoes on a hard surface as opposed to a structural defect that causes the floor to creak or snap under the weight of foot traffic. I think you are on the right track here. If you want to stay with a hardwood type floor, as opposed to carpet or vinyl, There are several sound rated underlayments for use under engineered hardwoods. Please, don't say the word laminate flooring or we will not be friends!!! It is not unheard of to use two layers of the closed cell foam backers. An alternative that might be a good compromise solution is vinyl plank or foam backed vinyl flooring. There are some very good looking vinyl products that look very much like wood or tile. I have been using a foam backed vinyl sheet product from Tarkett in many of my apartments that looks so much like real tile, folks have to get down on their knees to check it out. The nice thing is that my cost is only around $1.00 per square foot, a bargain. Some of the new vinyl plank products look more like real wood that real wood does! The last suggestion is to ban shoes in the house and issue soft fuzzy slippers to all residents!

You should be able to remove the carpet and padding without disturbing the tile. Then you can install a floating laminate wood floor without any concern.

A hardwood floor that requires nailing would be a bigger concern.