Question: I am in this, rather friendly, situation:

I have a complex projective space $\mathbb{P}^n$, and there i have a (possibly non-smooth) hypersurface $S$ defined by one irreducible polynomial $P$ of degree $d$.

What i want is to get information about the existence or not of linear subvarieties of $S$, and their maximal dimension $m$.
I seem to remember the existence of some ways to get bounds on $m$, given $n$ and $d$, but i don't remember anymore and i don't know where to look..

Answer: 
I think the key word you are looking for is Fano schemes of $S$. See this 
[note](http://www.math.sunysb.edu/~jstarr/papers/appendix3.pdf) by Jason Starr for a reference. For example, if $S$ is smooth and non-degenerate, it can not contain a linear subspace of dimension bigger than half $dim \ S$.

A very interesting related question is when can you find subvarieties of $S$ which was spanned in the Chow group by images of linear subspaces in $\mathbb P^n$. It was discussed in [this paper](http://www.math.neu.edu/%7Elevine/publ/SmallDegree.pdf) by Levine, Esnault and Viehweg.
I agree with Hailong, you want to look at Fano schemes. [Here](http://arxiv.org/abs/alg-geom/9611033)'s an article by Debarre and Manivel. It's very good, and I've gotten a lot out of it despite note knowing much French.