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In my experience SW trains are fine for bikes. As others have said they usually have dedicated spaces for a few, and if they're full you can stand with it by a door with no problem at all, never met a guard on SW trains who's complained, and that includes a few trips with about 10 of is with bikes dotted about.
If you're in the New Forest, make sure to pop into The Woods in Lyndhurst, they have a good cafe there now too.
thank you @spinnnout and @Jameo ! going sunday morning and back mid week - will try to get there extra early :eyes: