Abstract We study properties of Wilder, strongly Wilder, continuumwise Wilder, D , $$D^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> , and $$D^{**}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> </mml:math> Hausdorff continua. We present an example of a colocally connected continuum that is not a $$D^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>D</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -continuum, answering a question by Espinoza and Matsuhashi. We give several positive answers to this question for unicoherent continua. We also present some equivalences for the class of homogeneous Hausdorff continua with the property of Kelley.