Transactions spending one or more Segwit outputs are considered Segwit transactions.
BTC has Segwit since August 2017.
The main reason for the fork on 1 August 2017 was that BCH didn't want
Segwit (and still doesn't want it). They wanted to go on mining with their covert ASICboost. Segwit prevents this.
Segwit means: "Segregated Witness (Consensus Layer)" and it alleviates the scaling problem.
It was activated on August, 23rd 2017. In October 2017 Segwit network transactions rose from 7% to 10%.
During the last 3 days (7-8-9 October 2018), the percentage of Segwit-spending payments reached a new high. For the first time in history, the 7 day average was above 48%.
This statistic comes from aggregating all payments from Segwit transactions per day and comparing them to the number of total payments per day.
If your wallet doesn't support Segwit, well...it is time to find another wallet that does it.
If you don't want to change your wallet, then make them know Segwit is important to you by writing to their helpdesk or in every different way.
A partial list of wallets supporting Segwit:
Ledger Nano S (Hardware wallet)
Trezor (Hardware wallet)
GreenAddress (Web wallet and iOS wallet)
Electrum (Desktop wallet and Android wallet)
Armory (Desktop wallet)
Edge (iOS wallet)
BitWallet (iOS wallet)
Samourai (Android wallet)
GreenBits (Android wallet)