After Carroll, 1988, and Kearney & Stuart, 2004, with Recent species from Uetz, Etzold, & Chenna, 1995-2000 [“The Reptile Database”]

<==o Amphisbaenia Gray, 1844 [Annulata] (matoliskot, wormlizards)
   |?- †Amphisbaenia indet. Augé, 2003; L.Eoc. WEu.
   |?- †Omoiotyphlops de Rochebrune, 1884 (nomen dubium)
   |?- †Anniealexandria Smith, 2009
   |?- †Blanosaurus primeocaenus Folie, Smith & Smith, 2013
   |?- †Campinosaurus woutersi Augé, 1992; E.Eoc. WEu.
   |?-
   |?- †Cryptolacerta hassiaca Müller, Hipsley, Head, Kardjilov, Hilger, Wuttke & Reisz, 2011; M.Eoc. CEu.
   |?-o †Hyporhina Baur, 1893 [Hyporhinidae Baur, 1893]
   |  |-- †H. antiqua Baur, 1893
   |  |-- †H. galbreathi Taylor, 1951
   |  `-- †H. tertia Berman, 1972
   |?- †Louisamphisbaena Augé, 2012
   |?- †Oligodontosaurus wyomingensis Gilmore, 1942 [Oligodontosauridae Estes, 1975]
   |?- †Palaeoblanus Schleich, 1988; L.Olig.-E.Mioc. CEu.
   |?-o †Polyodontobaenidae Folie, Smith & Smith, 2013
   |  |-- †Camptognathosaurus parisiensis Folie, Smith & Smith, 2013; L.Pal. WEu.
   |  `-- †Polyodontobaena belgica Folie, Smith & Smith, 2013; E.Pal. WEu.
   `--+--o Rhineuridae Vanzolini, 1951
      |  |-- †Plesiorhineura tsentasi Sullivan, 1985
      |  |-- †Jepsibaena minor Vanzolini, 1951
      |  |--o †Ototriton Loomis, 1919
      |  |  |-- †O. minor Gilmore & Jepsen, 1945
      |  |  `-- †O. solidus Loomis, 1919
      |  |--o †Spathorhynchus Berman, 1973
      |  |  |-- †S. fossorium Berman, 1973
      |  |  `-- †S. natronicus Berman, 1977
      |  |-- †Lestophis crassus (Marsh, 1871) Marsh, 1885 [Limnophis crassus Marsh, 1871]
      |  |-- †Dyticonastis rensbergeri Berman, 1976
      |  |-- †Macrorhineura skinneri Macdonald, 1970
      |  `--o Rhineura Cope, 1861 [Gilmoreia Tayor 1951; Pseudorhineura Vanzolini 1951; Platyrhachis Cope, 1873]
      |     |-- †R. hatcherii Baur, 1893 [Gilmoreia attenuatus Taylor, 1951; Platyrhachis coloradoensis Cope, 1873; Pseudorhineura minuta Gilmore, 1938, Rhineura amblyceps Taylor, 1951; Rhineura attenuatus Taylor, 1951; Rhineura coloradoensis Cope, 1873; Rhineura hibbardi Taylor, 1951; Rhineura minutus Gilmore, 1938; Rhineura sternbergii Walker, 1932; Rhineura wilsoni Taylor, 1951]
      |     |-- †R. marslandensis Yatkola, 1976
      |     `-- R. floridanum (Baird, 1858)
      `--+--o Blanidae Kearney, 2003
         |  |-- †Cuvierbaena carlgansi Cernanský, Augé & Rage, 2015
         |  `--o Blanus Wagler, 1830
         |     |-- B. cinereus (Vandelli, 1797) (Iberian worm lizard)
         |     |-- B. mariae Albert & Fernández, 2009
         |     |-- B. mettetali Bons, 1963 (Moroccan worm lizard)
         |     |-- B. strauchi (Bedriaga, 1884) (Anatolian worm lizard)
         |     `-- B. tingitanus Busack, 1988
         `--+--o Bipedidae Taylor, 1951
            |  `--o Bipes Latreille, 1801 [] (“eturaajamatoliskot”)
            |     |-- B. biporus (Cope, 1894)
            |     |-- B. canaliculatus Bonnaterre, 1789 [B. alvarezi Smith & Smith, 1977]
            |     `-- B. tridactylus (Dugčs, 1894)
            `--+-- Amphisbaenidae Gray, 1844 (matoliskot)
               `--o Trogonophiidae Gray, 1865 [Trogonophidae; Trogonophididae] (vanhamatoliskot)
                  |--o Agamodon Peters, 1882
                  |  |-- A. anguliceps Peters, 1882
                  |  |-- A. arabicus Anderson, 1901
                  |  `-- A. compressus Mocquard, 1888
                  |-- Diplometopon zarudnyi Nikolsky, 1907 [Diplometopodon]
                  |-- Pachycalamus brevis Günther, 1881 
                  `--o Trogonophis wiegmanni Kaup, 1830 (matolisko)
                     |-- T. w. wiegmanni (Kaup, 1830)
                     `-- T. w. elegans (Gervais, 1835)

After Cernanský, Augé & Rage, 2015 (reduced to families)

<==o Amphisbaenia
   |-- Rhineuridae
   `--+--+-- Bipedidae
      |  `-- Blanidae
      `--+-- Trogonophiidae
         `-- Amphisbaenidae

Reference(s):

  • Cernanský, A., Augé, M. l. & Rage, J.-C., 2015: A complete mandible of a new amphisbaenian reptile (Squamata, Amphisbaenia) from the late middle eocene (Bartonian, Mp 16) of France.
    –Journal of Vertebrate Paleontology: Vol. 35, #1, pp. e902379 (1-9) [doi: 10.1080/02724634.2014.902379]

Reference(s):