As per Ron Maimon's suggestion, I did a little simulation for the lattice version, counting the number of paths
that get trapped at or before $n$ steps.
For example, here is a trapped path of 45 steps:
<br />![Trapped Path][1]<br />
Because there is a positive probability at any point of forming a shape like the letter 'G'
(if there is sufficient room), the probability of trapping goes to 1 as $n \rightarrow \infty$.
For $n=100$, the probability is already over $\frac{3}{4}$.
<br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;![Trapping Probability][2]<br />
<b>Addendum.</b> Incidentally, I have some evidence&mdash;not definitive&mdash;that the mean path length before 
reaching a _cul-de-sac_ is about 71.6 steps.


  [1]: https://i.sstatic.net/wyieb.jpg
  [2]: https://i.sstatic.net/pF9Ep.jpg