What greetings or sayings are there that are used during the new year, specific to the Year of the Snake?
You can search &quot;蛇年祝福语&quot; online, but most of the results are artificial and aren't as classy as the 龙 ones. If you look up for Chengyu's that contain 蛇, you'll see the majority of them are negative, due to snake's symbolic meaning in Chinese culture. Even novelists complained that it's very difficult to come up with greeting lines with 蛇.
The last link contains some examples of greeting phrases:

　　从新春祝词的角度，如“龙蛇飞舞”、“笔走龙蛇”、“灵蛇之珠”等寓意美好的成语也可一用。网友“felix77”还创作出了不少祝福语：“龙蛇共舞三春景、蛇舞九州处处新、蛇步锦程业辉煌、蛇游大泽燕游春、蛇含瑞气报年华 、蛇出山穴喜报春、蛇衔瑞草户迎春。”
　　如果实在是想不到跟蛇相关的吉祥话，还有网友剑走偏锋：“只要把蛇称作小龙，一切问题都迎刃而解……”不过，最简单的方法还是用一些万能格式，如“蛇年吉祥、蛇年大吉、金蛇献瑞”。还有网友想出“硬上弓”的法子：“祝新年蛇腾虎跃，蛇飞凤舞，蛇蛇生威，蛇到成功，蛇马精神……”

To explain a few:

龙蛇飞舞 (dragons and snakes are flying): Originally it's used to describe elegant and swift movement in calligraphy. When used as new year greetings, it's blessing the audience an elegant and swift conduct of life/career/business etc.

笔走龙蛇 (dragons and snakes are following one's writing brush): Original meaning is the same as above. Since 笔 (pen) is explicit in this one, it's more appropriate to use it against someone who indeed writes.

灵蛇之珠 (pearl of the snake): Invaluable treasure or unique talent.

蛇舞九州处处新: Snakes are dancing all over China, taking everything to a new start.

蛇步锦程业辉煌: Snakes are stepping towards to a bright future and making outstanding achievements.