Are there any existing codecs or modes, or are hams experimenting with
modes that provide better than telephone/radio quality sound?
Here in Canada it's pretty much only D-Star you hear about. Some hams in USA are experiencing with P25 I think, but only in conventional mode (no trunking). Both technologies use vocoders from DVSI inc, respectively AMBE and IMBE/AMBE+2 chips. I think right know Yaesu is coming with a new digital technology (12.5 kHz C4FM FDMA modulation) which looks like a P25 ham adaptation. According to this post, they are using AMBE+2 too. D-Star sound quality is terrible in comparison of P25. P25 sounds a bit better than telephone when properly configured but lesser than analog FM or radio (you're talking about AM/FM radio I guess?), especially with background noises (sirens, music, chatter, etc).
Are the newer digital modes Icom and others coming out with better
than telephone/radio quality?
None that I'm aware of. I work with D-Star and P25 and Tetra. The two latters having the best quality/intelligibility to my ear. D-Star being to worst. The use of proprietary technology (DVSI vocoders) in D-Star and Yaesu's digital gear makes many radio amateurs angry because it's not aligned with the "openness" philosphy shared in ham communities.
I know there is a new codec, open source, named Opus that could be used in ham radios, but I don't know if anyone checked it seriously as an alternative to DVSI proprietary vocoders.