The only ones I've noticed to use their Great House names at the beginning of their names are those who are in significant positions of power - Hlaalu Helseth, Hlaalu Athyn Llethan, Indoril Nerevar (all three kings), Indoril Brindisi Dorom (Duke of Mournhold), etc. Note that, for instance, the Councilors of the Great Houses don't seem to use the House names at all - just look at the various House books of 3E 426, where the only one with a House name is Hlaalu Athyn Llethan. House Dagoth seems to spread that name much more liberally throughout its leadership, but it's not a part of the main House system anymore, either.
Interestingly, Hlaalu seems to be a moderately common second-name surname among Dunmer from nobles all the way down to glass miners, but the other Great House names aren't - the only Dres in Vvardenfell is a member of House Telvanni, oddly enough, and there isn't anyone in any game named Redoran, Telvanni, or Indoril at all. Judging by that, the system seems to be that those who have the name by birthright put it at the end, but those who are major figures in Morrowind and are members of a Great House can put it at the beginning. Whether Helseth is a Hlaalu by birth and the second rule simply overrides the first, I don't know, but that would make sense to me and explain the lack of any other surname of his (and possibly his parents Barenziah and Symmachus, since they were Queen and consort of Morrowind themselves, and his sister Morgiah). Mostly speculation on my part, though.
Members of Great Houses would use their surname as a suffix,
Except, as I mentioned, there's a Dres in House Telvanni, and also a Hlaalu in House Indoril (presumably, as an Ordinator)... it doesn't seem to match up quite that well.