TMF has also been used to solve classical topological problems. For example Bruner, Davis and Mahowald obtained new results regarding nonimmersions of real projective spaces in Euclidean space (http://hopf.math.purdue.edu//Bruner-Davis-Mahowald/eo2.pdf).

One very exciting sounding paper (not yet available) is by Behrens, Hopkins, Hill and Mahowald:

We completely determine the image of the Hurewicz homomorphism for tmf. We draw certain conclusions on which dimensions can contain exotic spheres. In particular, the only dimensions below 126 which contain no exotic spheres are 1,2,3,5, 6, 12, 61, and perhaps 4.

Some slides for a talk about this is available here: http://math.mit.edu/~mbehrens/presentations/Exotic_spheres.pdf

I don't know if it falls into classical topology, but the calculations of tmf have been used to find minimal $v_2$-periodic self maps (for example http://math.mit.edu/~mbehrens/papers/v2_32.pdf).

I know you asked about topological problems, but it seems remiss to not points out that there are also connections to number theory. Perhaps a good starting point is the ICM address of Hopkins: http://arxiv.org/pdf/math/0212397v1.pdf

From the mathscinet review:

The article describes how the study of tmf relates the Hopf fibration to the Weierstrass ℘-function, and leads to a proof of a theorem of Borcherds on congruences for modular forms arising from $\theta$-functions associated to lattices. It discusses the relationship between tmf and the theory of p-adic forms of Serre and Katz. It explains that tmf is the natural receptacle for the Witten genus, hinting at an intimate and still incompletely understood relationship to string theory.

Another paper I'm fond of is a paper of Mark Behrens: http://math.mit.edu/~mbehrens/papers/betagt.pdf He associates to each additive generator of the 2 line of the Adams-Novikov spectral sequence a certain modular form. This is akin to the connection between the Image of J and Bernoulli numbers.

I'm sure I'm just scratching the surface here.

TMF has been used to solve classical topological problems. For example Bruner, Davis and Mahowald obtained new results regarding nonimmersions of real projective spaces in Euclidean space (http://hopf.math.purdue.edu//Bruner-Davis-Mahowald/eo2.pdf).

One very exciting sounding paper (not yet available) is by Behrens, Hopkins, Hill and Mahowald:

We completely determine the image of the Hurewicz homomorphism for tmf. We draw certain conclusions on which dimensions can contain exotic spheres. In particular, the only dimensions below 126 which contain no exotic spheres are 1,2,3,5, 6, 12, 61, and perhaps 4.

Some slides for a talk about this is available here: http://math.mit.edu/~mbehrens/presentations/Exotic_spheres.pdf

I don't know if it falls into classical topology, but the calculations of tmf have been used to find minimal $v_2$-periodic self maps (for example http://math.mit.edu/~mbehrens/papers/v2_32.pdf).

I know you asked about topological problems, but it seems remiss to not points out that there are also connections to number theory. Perhaps a good starting point is the ICM address of Hopkins: http://arxiv.org/pdf/math/0212397v1.pdf

From the mathscinet review:

The article describes how the study of tmf relates the Hopf fibration to the Weierstrass ℘-function, and leads to a proof of a theorem of Borcherds on congruences for modular forms arising from $\theta$-functions associated to lattices. It discusses the relationship between tmf and the theory of p-adic forms of Serre and Katz. It explains that tmf is the natural receptacle for the Witten genus, hinting at an intimate and still incompletely understood relationship to string theory.

Another paper I'm fond of is a paper of Mark Behrens: http://math.mit.edu/~mbehrens/papers/betagt.pdf He associates to each additive generator of the 2 line of the Adams-Novikov spectral sequence a certain modular form. This is akin to the connection between the Image of J and Bernoulli numbers.

I'm sure I'm just scratching the surface here.