I'm curious, what is the significance of using a wallet that is supporting segwit?
Will it have additional features?
Will my old private keys work on it?
Pardon the ignorance, I haven't really invested time in reading and knowing segwit.
To actually be able to create SegWit transactions, you first have to move your coins to a SegWit (P2SH) address. Those start with a "3", like multi-sig addresses, and being separate addresses will have private keys of their own (although with hardware wallets / deterministic wallets your seed will stay the same).
SegWit transactions will enable you to send transaction for lower fees, compared to legacy transactions. Further down the road it will be also enable you to use 2nd Layer solutions such as Lightning Network, bringing down transaction fees even more.
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@OP, bitcoin unlimited and bitcoinABC probably support it as they are based on the source of bitcoin core and I see no reason for them to remove that code.
Nope. Bitcoin unlimited and BitcoinABC got forked with the main purpose of removing SegWit, so those wallets definitely don't support it.