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There’s been a bunch of motorbike threads on here recently so I thought I’d ask.
Basically, all my commuting options are grim (current best option is get a lift and be an hour early every day) and I’ve ALWAYS wanted a bike. Yes, a big bike is a million times nicer to ride than a 125 and I’ve got a list of big bikes I’d rather have but I don’t have a licence.
If I go the CBT route I could be commuting to work on a brand new bike by next week. Plus I’d be filtering through stop/start traffic maxing out at 40mph if I’m lucky so massive torques are not essential. If I wake up and the weather is horrible I’ll jump in the car with the Mrs like I do now. It would also give me the incentive to work towards a full licence while enjoying gentle pootles around the Peak District.
I know 125’s aren’t exciting but I really like the Mutt. I know it’s a try-hard, fake classic, but I prefer that to a plastic sports bike that sounds like a strimmer. It’s no bigger than my mountain bike, it’s cheaper (especially if I can find a rare used one) and it’s my favourite colour. I’ve already got a helmet and boots.
What’s not to like?

Second hand CG125 would probably be a better option. Couple of cans of BBQ paint and a pair of chunky tyres and you're all set.
No experience of the Mutt but another vote for CG125, CBF125 or a Varadero 125 for a bit more weather protection
I have had big bikes all my life, but for fun in restricted places like cities or narrow roads, it's hard to beat a small bike.
So don't let size put you off. The important thing on a small bike is its suspension. It's got to be good enough to keep the wheels on the ground - I've found the quality of damping suspect on some. Tuning them is also fun, but start with brakes, tyres and suspension, before doing anything to the engine.
I'd buy a Skyteam Ace if it was my choice. People are popping 300cc engines in them 🙂

They look cool, will hold residual well if you can look after it and stop it rusting.
Sounds like a commute where I would rather use a 125 than my big bike.
There's a few similar companies that offer cheap Chinese bikes that have been modded for life in the UK. Herald are pretty similar to Mutt (www.heraldmotorcompany.com) and Sinnis have a few options.
I went and sat on a Mongrel after work and also a "Fat Sabbath". The fatty looks great but I can imagine those knobbly tyres being a bit of a drag on a feeble engine.
They feel like solid bits of kit to be fair. I was trying to look for nasty welding or whatever shonky workmanship was hiding behind the cool paintwork but they're pretty convincing up close. All the contact points are fitted in England and feel good quality. They don't have giant, dorky orange indicators or those massive plastic brake light housings.
I'm no expert but I'm sure a proper bike nerd could point out the shortcomings. It felt about the same size as my Geometron and I bet the wheelbase isn't far off!
They also had a Varadero in, painted in a strange shade. It looked a bit...

People are popping 300cc engines in them
Didn't want to get into cheeky engine swaps but there's a 250 version of the Suzuki rip-off in the Mutt that slips straight in. Definitely something to think about in the future if you liked the bike and didn't want the hassle of buying and selling.