I recently started learning a little model theory and in particular I found [this][1] nice exercise.  I only know a little topology, but this prompted me to wonder how many model category structures there may be on Top.  I am aware of three: Serre fibrations and weak homotopy equivalences, Hurewicz fibrations and homotopy equivalences, and the usual model category of rational homotopy theory [[Dwyer and Spalinski]][2]. A secondary question could be how many homotopy theories there are since it is known that the first two I mention give the same homotopy theory.

  [1]: http://mathoverflow.net/questions/29635/what-determines-a-model-structure/29653#29653
  [2]: http://folk.uio.no/paularne/SUPh05/DS.pdf

*This question is a little out of my league right now. I hope that is ok. I'm not even sure how difficult this question is.