There is an impossibility theorem: If you let $\mathcal{L}$ be the the space of Borel-measurable functions $f:[0,1]\to[0,1]$, and $e:\mathcal{L}\times [0,1]\to[0,1]$ the evaluation given by $e(f,x)\mapsto f(x)$, then there is no $\sigma$-algebra on $\mathcal{L}$ such that the evaluation is jointly measurable. The result is a consequence of the rather complicated classification result in  R. Aumann, [Borel Structures for Function Spaces][1], Illinois Journal of Mathematics 5 (1961), pp. 614-630. Easier proofs of the main results can be found in the paper "Borel Structures for Function Spaces" (yes, same title) by B.V. Rao, Colloquium Mathematicum, 1971.

A $\sigma$-algebra on measurable functions I have actually seen used is the following: If $(S,\mathcal{S})$ and $(T,\mathcal{T})$ are measurable spaces, we endow the family of measurable functions between them with the $\sigma$-algebra generated by sets of the form $\{f:f(s)\in B\}$ with $s\in S$ and $B\in\mathcal{T}$. The author used this $\sigma$-algebra to show that to each Markov kernel from $S$ to $T$, there corresponds a certain probability measure on this $\sigma$-algebra. The paper is H. v. Weizsäcker [Zur Gleichwertigkeit zweier Arten der Randomisierung][2], Manuscripta Mathematica 11 (1974).


  [1]: http://11.%20%22Borel%20Structures%20for%20Function%20Spaces,%22%20Illinois%20Journal%20of%20Mathematics%205%20(1961),%20pp.%20614-630.%20%20/
  [2]: http://www.digizeitschriften.de/dms/img/?PPN=PPN365956996_0011&DMDID=dmdlog9