Preventing staining of clothes when carrying bike
It might be a facetious answer on the face, but seriously - why are you carrying your bike at all?
Do carry a working mini-tube, pump, two tyre levers and whatever you need to get the wheel completely off if its not a QR (ring spanner, perhaps allen key, maybe special tool for IGHs etc)   Some people like disposable gloves for the hand protection.   Replace the tube, clear the cause, and ride on.  Remember to patch the tube at home, or replace it if its dead.

If you truely have to walk it is to simply wheel it.  Take extra care rolling over kerbs/curbs or any other edges.  The sidewalls of your tyre will be okay for a bit as long as you're not riding it.  Any road bike under 15 kg with clincher tyres will be okay rolling on the tyre for 5 km.

You've done something where it just won't roll.  If its the front, haul the front wheel off the ground in a vertical wheelie and roll it on the rear.  Hold the handlebars and use the rear brake if you need to.
3b. If its the rear that is wedged (and this is much more likely) then slip right hand fingers through the top of the wheel and carry it just off the ground.  Use your left hand for steering or put your left hand on the saddle and push it with &quot;no-hands&quot; steering   You're on the bike's left away from the chain.

Both wheels are buggered, and it won't roll at all?  Cyclocross carry, which is either like this:



or with the shoulder further back in the triangle, and the right hand on the fork or wheel rim or through to the bars.  These ones needs shoulder padding pretty quick, so whip out something cloth or a newspaper.

Note all carrying should be done from the non-drive side of the bike.  If you get sore, swap sides but spin the bike as well so you're still on the non-drive side and now the bike is facing backward.