Public transportation in Nepal

The buses are old and the taxi seats completely worn flat.
For a 3rd world country I would venture to say the transportation system is very well organized, may not be anywhere near as fast as you might prefer but what do you expect for 10-15 USD, from one large city to the next.
Most buses leave on time and arrive early, thanks to the driver taking corners fast and stopping for every didi along the way.
Relatively safe yes from the stand point of single females may be watched by every male on the bus, most of the trip but there is little fear of being assulted while traveling on public transportation. From the stand point of mechanically safe, just don't expect any mode of transportation to have a current inspection sticker on it.
I prefer to walk while in KTM, saves rupees and the roads are so congested you get a better feel for the city on foot and often make better time then taxis or rickshaws, as they have to wait for the 85 other taxis in front of them you can just walk past all of them.
The main roads of most citys have local bus service. Mini buses (5-20 rupees depending on your destination) can be caught on nearly every heavily trafficed road, most travel to major destinations-bus parks, city center, temples...

I prefer the local bus for city to city travel. mostly because I have paid 'tourist price' on a 'tourist bus' and it was the same bus. So avoid the misconception that you are getting a better standard of bus. Pay the local bus price and arrive at the same time at your destination with a few more rupees in hand. You can always buy an extra lassi or beer to reward yourself for not being duped.