Is an infinite incomputable sequence random wrt a computable measure?
Mucknik, Semenov and Uspensky showed that there are sequences which are not Martin-Löf random for any computable measure.  They call all other sequences (which are Martin-Löf random for some computable measure) "natural sequences".


  Andrei A. Muchnik, Alexei Semenov, and Vladimir Uspensky. Mathematical metaphysics of randomness. Theoretical Computer Science, 207(2):263–317, 1998.


This topic is also discussed in 


  Laurent Bienvenu and Christopher Porter. Effective randomness, strong reductions and Demuth’s theorem.
  
  Jan Reimann and Theodore A. Slaman. Probability measures and effective randomness.
  
  Jan Reimann and Theodore A. Slaman. Measures and their random reals.


Here is a quick proof (which may be different from the above references):  

Let $\{U_i\}$ be a (non-effective) enumeration of all $\Sigma^0_1$ sets making up some level of the universal Martin-Löf test for some computable measure $\mu$.  Each $U_i$ is dense and open, so by the Baire category theorem (countable intersection of dense open sets is dense), the intersection $\bigcap_i U_i$ is nonempty.  Any sequence in this intersection cannot be Martin-Löf random for any computable measure $\mu$.

The same can be done for Schnorr randomness (which is weaker than Martin-Löf randomness). Let $\{U_i\}$ be the collection of all dense $\Sigma^0_1$ sets of computable $\mu$-measure for some computable measure $\mu$.

UPDATE Oct 23, 2013:

I realize there are three things to add (which I am sure are all well-known and they are probably in the above mentioned papers).


My above proof actually shows that 1-generics cannot be Martin-Löf random for a computable measure, since 1-generics are in all dense $\Sigma^0_1$ sets.
The K-trivials are not Martin-Löf random on any computable measure.  A sequence $X \in \{0,1\}^\mathbb{N}$ is low-for-random (a.k.a. $K$-trivial) if $\text{MLR}^X = \text{MLR}$. It can be shown that this property also extends to all computable $\mu$.  That is, if $X$ is K-trivial, then $\text{MLR}_{\mu}^X = \text{MLR}_{\mu}$ for all computable $\mu$. Hence, the non-computable $K$-trivials cannot be random for any computable measure.  For if a $K$-trivial $X$ is random on a computable measure $\mu$ then $X$ is random relative to $X$.  This means $X$ is an atom.  However, only computable points can be atoms of computable measures.
The set of all "natural sequences" is precisely the set of sequences truth-table reducible to a Martin-Löf random (on the fair-coin measure).  (The main idea is that every computable measure is the push-forward on an a.e. computable map, and the Martin-Löf randoms on the measure $\mu$ are exactly the sequences pushed over to the new measure.  The reason this computation is "truth-table" involves using "layerwise computability".)