XXY (Klinefelter's) & Trisomy 21 & X-inactivation
X-inactivation does occur in XXY individuals. 

In phenotypically normal XX females, one X chromosome is inactivated.  Which one of the two X chromosomes is inactivated in any given cell is random. Therefore, about one half of the cells should have the maternal X active and the other half should have the paternal X active.

XXY individuals show a wide range of clinical symptoms and intellectual ability.  Iitsuka et al. (2000) suggested that this variation may be explained by a bias in X-inactivation.   They found that about 31% (5 of 16) individuals (either XXY or XXYY) shows a very strong bias of X inactivation, which they defined as at least 80% of the cells inactivating one particular X chromosome. Two individuals has biased inactivation of the paternal X and the other three had biased inactivation of one of the maternal X chromosomes. The sample size is rather small (not surprising) but X inactivation is occuring.  Samango-Sprouse (2001) argues that incomplete inactivation of one X chromosome may also explain some of the more severe clinical mental disorders.

For Trisomy 21 (Down's Syndrome), the individual as three copies of chromosome 21, so X inactivation is not an issue. However, scientists are studying whether Xist can shut down the third copy of chromosome 21.  The Xist gene is transcribed to produce a large molecule of RNA. That RNA is what usually causes X inactivation.  Jiang et al. (2013) were able to show that Xist can coat one of the chromosome 21 copies, performing essentially the same inactivation function that it does for the X chromosome. So far, though, this work has been entirely in the lab (in vitro).