There is growing interest in revealing exotic properties of collective spin excitations in kagome-lattice ferromagnets such as magnon Hall effects,topological magnon insulators,and flat magnon bands.Using the well-established nearest-neighbor Heisenberg ferromagnet model with Dzyaloshinskii-Moriya interaction (DMI),in this study we uncover intriguing new aspects in the selectivity and topology of fiat magnon bands.Among the three magnon bands (except for the top one,which is fiat in the absence of DMI),we observe that each of the three bands can be selectively flattened at the critical DMI of D =± √3J/3 and D =± 3J.With a general DMI,the magnon bands become non-flat;however,there are nested lines that create a David star pattern for all three magnon bands whose flatness is robust during changing exchange coupling or DMIs.Contrary to prevailing belief,we show that each of the three flat bands is actually topologically trivial at critical DMIs.Furthermore,we show that while the middle band remains topologically trivial,for the other two bands,D =0 corresponds to the topological phase transition where their Chern numbers get interchanged;when D =± √3J,the system undergoes a phase transition to the nonferromagnetic state.These central findings increase our understanding of spin excitations for future magnonics applications.