Dear Welshbear
As you have got the 1901 Census and working my way back to 1851, I found the following:
1891 Census - William & Martha MORGANS (s on end) with 6 children at 2 Bigyn Lane, Llanelli, Carmarthenshire
RG12 Piece 4494 Folio 96 Pg 7
1882 - their eldest son Thomas John MORGAN born March Qtr 1882 at Llanelli, Carmarthenshire Vol 11a Pg 751
1881 - Marriage of William MORGANS & Martha RICHARDS Sep Qtr 1881 at Llanelli Carmarthenshire Vol 11a Pg 953
1881 Census - Martha RICHARDS poss a servant with David Griffiths Grocer & Dealer at 22 Mount Pleasant Llanelli
RG11 Piece 5374 Folio 79 Pg 18
Martha's mother re-married and was shown as Hannah JONES (widow) and living with her in 1881 was Martha's full brother John RICHARDS and three half brothers at Tyrfran Cottages, Llanelly
RG11 Piece 5374 Folio 92 Pg 9
1871 Census - Martha RICHARDS was aged 11 and living with her step father (Thomas JONES) and mother (Hannah JONES). Martha RICHARDS is shown as being a step daughter to the head of the family (as is John RICHARDS shown as step son and Anne RICHARDS step daughter). They were living at Tyrfran Llanelly
RG10 Piece 5467 Folio 58 Pg 26
1861 Census - Martha RICHARDS aged 1 and living with her widowed mother Hannah RICHARDS along with Martha's 3 sisters & one brother. They were living at Cwmddiche Fach, Furnace, Llanelly Carmarthenshire
RG09 Piece 4114 Folio 47 Pg 2
1851 Census - Thomas RICHARD (head) & Hannah RICHARD (wife) and Elizabeth RICHARD aged 6 months
living at Furnace, Llanelly HO107 Piece 2468 Folio 311 Pg 52
On every Census all born Llanelli / Llanelly (both spellings are correct). Sometimes the surnames were RICHARD / RICHARDS and MORGAN / MORGANS.
I hope the above helps a little.
Regards, Alltcafan