Runes are primarily designed for being carved in wood (which is why there are so many vertical, straight lines, but no horizontal lines and very few curved ones). Wood does not survive very well, but sometimes it reaches us, there have been huge finds of pieces carved with the most mundane things - messages from telling people to go home, prayers and love letters (see e.g. Bryggen inscriptions). Thus, runestones were never the only or most common useage - they are just what has survived best. There are many ideas of why there were a wave of runestone's starting in Denmark with Harald Bluetooth and moving into Sweden and then continuing for several decades in the area around modern Stockholm, but it never reached Norway or Iceland (still, there are some Norwegian runestones, but not as many as in Sweden or even Denmark). It is possibly linked to the conversion to Christianity in some way, as the majority of stones with clearly religious symbols has them in the form of a cross.
Runes were certainly known to the settlers on Iceland - Egil's saga contains a passage about a boy trying to use rune magic to make a girl fall in love with him, but so unskilled made that it made her sick. Egil finds the reason and cures her. Rune magic is well attested also in other Icelandic manuscripts, and seems to have been the suriving useage on Iceland after the transition to latin script. Snorri Sturluson once recieved a message warning him of an conspiracy which was written with cipher runes. He either could not translate them or ignored the message, but at least it shows that knowledge of runes where not uncommon on Iceland in the middle of the thirteenth century. In some parts of Sweden, runes continued to be used for real text into the 18th and 19th centuries.
Finally, the "m" rune was used in manuscripts, both on Iceland and in mainland Scandinavia, as a shorthand for "man" well into medieval times.