We have been searching for surviving companions of progenitors of Galactic Type-Ia supernovae, in particular SN 1572 and SN 1006. These companion stars are expected to show peculiarities: (i) to be probably more luminous than the Sun, (ii) to have high radial velocity and proper motion, (iii) to be possibly enriched in metals from the SNIa ejecta, and (iv) to be located at the distance of the SNIa remnant. We have been characterizing possible candidate stars using high-resolution spectroscopic data taken at 10m-Keck and 8.2m-VLT facilities. We have identified a very promising candidate companion (Tycho G) for SN 1572 (see Ruiz-Lapuente et al. 2004; however for a different view see Kerzendorf et al., 2012) but we have not found any candidate companion for SN 1006, suggesting that SN event occurred in 1006 could have been the result of the merging of two white dwarfs (see González-Hernández et al., 2012). Adding these results to the evidence from the other direct searches, the clear minority of cases (20% or less) seem to disfavour the single-degenerate channel or that preferentially the single-degenerate escenario would involve main-sequence companions less massive than the Sun. Therefore, it appears to be very important to continue investigating these and other Galactic Type-Ia SNe such as the Johannes Kepler SN 1604.