How robust is discrete logarithm in $GF(2^n)$?
Discrete logarithms in $\mathbb{F}_{p}$  share the same asymptotic complexity as integer factorization for general numbers: $L_p[1/3,1.923]$ for general integers, $L_p[1/3,1.587]$ for special integers. Discrete logarithms in $\mathbb{F}_{p^n}$ have the same asymptotic complexity as factoring special integers, i.e. $L_{p^n}[1/3, 1.587]$, via the Function Field Sieve. 

So, extrapolating from the factoring records, we can handwave that a discrete log in $\mathbb{F}_{2^{1021}}$ is an order of magnitude easier than a discrete log modulo a 768-bit general prime, and about the same as modulo a pseudo-Mersenne prime $2^{1021} - c$.

As to whether composite degree extension fields are easier to solve, maybe. It is possible to represent the same composite field in a number of ways (usually known as tower of fields), and it is possible that some representations allow for faster breaks than others. Here's a quote from Andrew Odlyzko's 1985 paper Discrete logarithms and their cryptographic significance:


  In fact, these ﬁelds may be very weak because of the possibility of moving between the ﬁeld and its subﬁelds.


However, there is no data on significant asymptotic advantages of composite degree over prime degree (if there was, pairing-based schemes would be toast, as embedding degrees are by and large composite). 

One shouldn't also forget to check the smoothness of $p^n-1$, to avoid embarrassing Pohlig-Hellman breaks.

May 2013 Update

It seems that composite degree is indeed weaker than prime degree. The recent results of Göloğlu et al and Joux attack fields $\mathbb{F}_{2^n}$, for composite $n$, much faster than the function field sieve referred above. See this question for more information.