Can Bitcoin Diamond be claimed on Segwit addresses? I currently have my Bitcoin (BTC) on a Trezor hardware wallet with Segwit (Not in a "Legacy account").

Can I claim Bitcoin Diamond (BCD) with this or not?

In general: Should I move my BTC to a Legacy account to be more compatible with future forks?
Bitcoin Diamond does support segwit, so it is possible to claim it. However, it may not be possible to (safely) do it directly from the trezor.

There is apparently an Electrum for Bitcoin Diamond, which claims to support hardware wallets. However, I have never used it, and would be hesitant to trust it to not send the coins to its own address (maybe even from other chains).

If you really do want to claim it, I would suggest:


Move all coins from the trezor, for all chains, to a temp. wallet
Reset the trezor to get a new seed
Send all coins back
Use the old seed to derive your keys using a BIP39 tool, such as https://iancoleman.io/bip39/
Import the required keys into the core client for Bitcoin Diamond, and sweep them.