first of all, hi guys!
Officially the Korean and Japanese languages are isolated, they don't belong to any language category yet.
indeed it depends meanly from the linguistic "traditions", the russians (who know the altaic languages better that the occidentals) believe that the korean and japanese are altaic languages, the occidental linguists instead are usually more cautious (but generally they don't exclude it).
I'm think that the russian position is very plausible, and if you look the "etymological dictionary of the altaic languages" of S. Starostim, A. Dybo, O. Mudrak you will find not only a very large nomber of common roots, but a list of common suffix and grammatical structures too.
Some example (grammar):
proto altaic (PA) plural suffix *-t'- :
P-TungusManchu (TM) : *-ta(n) / *-te(n), Mong. -d, Turk *-t, PJapanese (PJ) *ta-ti, PKorean (PK) *-tï-r
PA accusative *be:
TM *ba/*be, OJ wo
PA partitive *ga:
TM *ga, OJ possessive ga, ?mong. accusative *-gh
PA genitive *-n´V:
PTM *ngi, J no, Kor -n´, Mong *n
PA 1st person pronoun bì
Tung. *bi, *bue, *mü-n-; Mong. *bi, *min-; *ba, *man- ; Turk *be- ; Jpn. *bà-; Kor. *úrí
and i could continue...
Some example of words, i open a page by chance:
PA *miûko "snake":
Tung. *mükü, Mong *mogaji, turk *böke, Jpn *múkátai, Kor *mëk-
PA *miuktu "stump", "horn fundation":
Tung. *mugde-ke, Mong. *mugdji-, Kor. *mìth
PA *miul´i "ignorant":
Tung. *mul-(di-), Mong. *mulgu-, Jpn. *misu-, Kor. *mòrá
PA *miúl´i "ice", "hail"
Mong. *möl-sü, *möl-dür, Turk. *bû(n)r´, Jpn. *mínsë-rá-, Kor. *múrúi
PA *miùní "a kinf of skin or cloth":
Tung. *mune-, Jpn. *mìnë
PA *miûndjù "a kind of badger":
Tung. *mô[ndj]ika, Mong. *mindji, Jpn. *mùnsìnà
PA *miungo "suffering":
Tung. *mingî-, Mong. *mung, Turk. *bung, Jpn. munkua-
...
And there are about 3000 roots in this dictionary...