Shape coexistence and multiparticle-multihole structures in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Cd</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>110</mml:mn><mml:mo>,</mml:mo><mml:mn>112</mml:mn></mml:mrow></mml:mmultiscripts></mml:math>

From detailed spectroscopy of $^{110}\mathrm{Cd}$ and $^{112}\mathrm{Cd}$ following the ${\ensuremath{\beta}}^{+}/\mathrm{EC}$ decay of $^{110,112}\mathrm{In}$ and the ${\ensuremath{\beta}}^{\ensuremath{-}}$ decay of $^{112}\mathrm{Ag}$, the presence of very weak decay branches from nonyrast states is revealed. In $^{112}\mathrm{Cd}$, ${2}_{5}^{+}\ensuremath{\rightarrow}{0}_{4}^{+}$ and ${4}_{6}^{+}\ensuremath{\rightarrow}{2}_{5}^{+}$ transitions are observed that yield $B(E2;{2}_{5}^{+}\ensuremath{\rightarrow}{0}_{4}^{+})=34\ifmmode\pm\else\textpm\fi{}15$ W.u. and $B(E2;{4}_{6}^{+}\ensuremath{\rightarrow}{2}_{5}^{+})=77\ifmmode\pm\else\textpm\fi{}30$ W.u., respectively, clearly indicating a collective structure. In $^{110}\mathrm{Cd}$, a weak decay branch from the ${4}_{6}^{+}$ level to the ${2}_{5}^{+}$ level is observed, and from a lifetime measurement following the $(n,{n}^{\ensuremath{'}}\ensuremath{\gamma})$ reaction, $B(E2;{4}_{6}^{+}\ensuremath{\rightarrow}{2}_{5}^{+})=55\ifmmode\pm\else\textpm\fi{}14$ W.u. is determined. A new branch is also observed for the decay of the ${6}_{4}^{+}$ level to the ${4}_{6}^{+}$ state, indicating that the sequence ${2}_{5}^{+}$, ${4}_{6}^{+}$, and ${6}_{4}^{+}$ forms part of a collective structure. The presence of ${3}_{3}^{+}$ and ${5}_{2}^{+}$ levels spaced between the previous sequence is highly suggestive of a $\ensuremath{\gamma}$ band built on the ${0}_{2}^{+}$ shape-coexisting intruder state. The ${0}_{4}^{+}$ levels in $^{110,112,114}\mathrm{Cd}$ have preferred decays to the lowest ${2}^{+}$ members of the intruder bands, and for $^{114}\mathrm{Cd}$ a previous measurement had established an enhanced $B(E2;{0}_{4}^{+}\ensuremath{\rightarrow}{2}_{3}^{+})$. The energy systematics of the ${0}_{2}^{+}$, ${0}_{3}^{+}$, and ${0}_{4}^{+}$ levels all display the characteristic parabolic-shaped pattern, suggesting that they are built on multiparticle-multihole proton excitations. The results are compared with beyond-mean-field calculations that reproduce qualitatively the observed levels and their decays and suggest that the ${0}_{1}^{+}$, ${0}_{2}^{+}$, ${0}_{3}^{+}$, and ${0}_{4}^{+}$ levels and the excited states built on them possess different deformations.