Has US real median personal/household income been stagnant since the late 1970's/early 1980's?
As said in the comments, there is an important difference between household income and wages. Household income includes income from capital. As a higher share of total income is going to capital than to labour (see below), you would expect a divergence between household and wages. 



Similarly, productivity increases are not being translated into higher wages, which means a higher distribution of gains toward capital (see below).



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Another important issue is that the median can be a misleading figure when distributional changes are occurring. For instance, below is the percentile level of household income for the US:

 

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As you can see, the median can mask quite an heterogeneous development for different percentiles. Since lower percentiles source their household income mainly from wages, a stable median wage but an increasing median income is entirely consistent with higher inequality plus lower labour share.