Update: It seems the fruit collecting levels are generally the most difficult, and are the ones that people are having trouble with. I have completed the first 50 fruit collecting missions (without using any powerups) and have posted the solutions here: Cut the Rope Fruit collecting missions walkthrough
Please see the youtube links to my Cut the Rope 2 video solutions below.
Currently I have solutions to most of the first 20 levels of Cut the Rope 2, using no powerups!
Some achievements I have found too difficult to complete without powerups, particularly the ones which involve collecting a certain amount of fruit. I will keep working on those ones, and will hopefully have a solution to every achievement soon!
Cut the Rope 2 - Forest - Level 1 - 3 Stars
Cut the Rope 2 - Forest - Level 2 - 3 Stars
Cut the Rope 2 - Forest - Level 3 - 3 Stars
Cut the Rope 2 - Forest - Level 4 - 3 Stars
Cut the Rope 2 - Forest - Level 5 - 3 Stars
Cut the Rope 2 - Forest - Level 6 - 3 stars
Cut the Rope 2 - Forest - Level 7 - 3 Star walkthrough
Cut the Rope 2 - Forest - Level 7 - Collect 30 fruit
Cut the Rope 2 - Forest - Level 7 - 0 stars
Cut the Rope 2 - Forest - Level 8 - 3 Star walkthrough
Cut the Rope 2 - Forest - Level 8 - Collect 25 fruit
Cut the Rope 2 - Forest - Level 8 - 2 stars
Cut the Rope 2 - Forest - Level 9 - 3 Star walkthrough
Cut the Rope 2 - Forest - Level 9 - Collect fruit
Cut the Rope 2 - Forest - Level 9 - 0 stars
Cut the Rope 2 - Forest - Level 10 - 3 Star walkthrough
Cut the Rope 2 - Forest - Level 10 - Collect 39 fruit
Cut the Rope 2 - Forest - Level 10 - 1 star