<<I agree with the suggestion of constructing one's own antenna, for non-mobile. But dipoles are less than optimal due to their directiveness.>>
So you hang it up as a vertical dipole, with the wire connected to the braid of the co-ax pointing downwards. That way it becomes omni-directional.
That sounds suspiciously like a clever disguise of the ground-plane antenna I recommended - but I am not an expert on antennae.
Be forewarned - a lot of cheapo scanners will become overloaded with a more efficient antenna. It won't damage the set, but you may well hear transmissions which you are not tuned to. Eg a taxi driving past, or the local CB or amateur transmitters may blast through well away from their transmit frequencies.
It was never a problem to me - neither on the Maycom AR108 (how's that for cheapo?) nor on the Icom A6. Perhaps an indication my ground-plane concoction is far from perfection? Or then, in this rural village taxi's are most rare, and CB won't perhaps be introduced for a couple of decades.
Sometimes the AR108 sounds messages that seem to come from high up - FL300-400 perhaps - I take these to be on 8.33 frequencies that slip through its 25 KHz bandwidth. This happens very rarely, though, even if there's plenty of long-distance flights high up over here.