Can a single measurement on an arbitrary qubit determine if it had been entangled or not? I have a single isolated qubit. Can I perform any measurement that will tell me if it had been entangled with another qubit or not?
 A: It depends on what exactly you mean.
If you only look at a single qubit, it is impossible to tell whether the qubit was entangled/correlated with something else. You don't need to restrict to a single measurement scenario here: measure as much as you like (which here means, measure any number of copies of the same qubit part of the same system), it won't make any difference. 
This is essentially the no-communication theorem. Given a bipartite state $\rho\in\mathcal H^{d_1}\otimes\mathcal H^{d_2}$, if you only measure/operate on the first part of the system (in your case, the qubit), your state is for all practical purposes $\rho^1\equiv\operatorname{Tr}_2(\rho)$, and there is no way to tell whether $\rho^1$ is correlated with something else$^1$.
This changes if, after measuring your system, you are allowed to compare measurement results with someone (say, Bob) that measured the other part of the system. In this case, if both of you use suitable measurement bases, you can figure out whether you and Bob shared an entangled system or not. A paradigmatical example of this kind of thing is you and Bob performing a Bell experiment to violate some kind of Bell inequality.
On the other hand, if the question is: "can I tell whether my qubit was entangled with something else, by performing a single measurement and comparing the result with someone holding the other part of the system?", then the matter becomes trickier.
How much information about the correlations can be probed by comparing local measurement on the two parties will strongly depend on many things, such as what assumptions on the state are made (if any), what kind of measurements are performed, whether you allow successive measurements to depend on the previous ones, etc.

$^1$ It is worth noting however that this is only true if no assumptions are made. If on the other hand you assume that the global state is pure, then local measurements can tell you whether $\rho^1$ is pure or mixed, and therefore whether it is correlated with the rest of the system.
