Open/modern question in labor economics? What are some of the modern lines of inquiry in labor economics and what are some good sources of them (especially, succinct ones)?
If you want a recent state of the art in Labor Economics, you can have a look at the last Handbook of Labor Economics (2011, edited by Card and Ashenfelter). Though it is getting old, it provides an overview of specific approaches or questions in Labor. It is especially relevant if you want to define a research project for a PhD.  

To be more specific, here is a (non-exhaustive) list of trendy research questions (you can use these keywords to find research articles and reviews):


Polarization of the labor market. There is evidence of a decline in demand for middle-skilled workers (less employement and reduce in wages). The two main explications are globalization and skill-biased techonological change. Researchers try to weight the role of each explanation from data. In particular, a recent approach interests in the type of tasks in each occupation. This is related to the chapter by Autor and Acemoglu in the Handbook. Jobs with routine tasks are likely to be affected by technology as they can be more easily automated. Looking at the type of tasks, whether it uses cognitive or non-cognitive skills, brings new insigths in other interesting questions (returns to education, gender/racial wage gaps, local labor markets). You can look at some work by David Autor or David Dorn for instance.
Productivity puzzle. In many advanced countries, productivity has not found its before-crisis trend, and we do not know why. Research focused in country-specific situations (the UK productivity puzzle, the French productivity puzzle, etc) to find the reasons of this productivity slowdown in the national labor markets. It is also a puzzle that so many countries are affected despite their differences in the functionning of their economy.
Labor market, household decisions and marriage market. A rich literature studied the labor supply choices within the household as a joint decision. A new research area relates labor market outcomes to the formation of households. The novelty is to consider that individuals are not matched in a pure randomly way on the marriage market, and decisions to marry may account for individual abilities and skills. In particular, it has been shown that high-skilled female workers marry high-skilled male workers more often than in the past, and reciprocally. This has consequences in terms of inequalities and labor supply decisions. There are recent works conducted by Pierre-André Chiappori among others.