Okay, you have to jump around in a couple of different places to see your options here.
From a sidebar in the back of the Monster Manual, (NPC section, p347) we get:
"Any spellcaster that can cast the find familiar spell (such as the archmage or mage) is likely to have a familiar. The familiar can be one of the creatures described in the spell (see the Player's Hanbdook) or some other Tiny monster, such as a crawling claw, imp, pseudodragon, or quasit."
From the Monster Manual listing of pseudodragon (p254, and similar text is found for imp and quasit, though not crawling claw):
"Mages often seek out pseudodragons, whose...make them superior familiars.
...
Some pseudodragons are willing to serve spellcasters as a familiar...
...
The pseudodragon can serve another creature as a familiar...
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At any time and for any reason, the pseudodragon can end its service as a familiar..."
Now, there are a few different ways you can interpret this.
One could be a jerk and say that since the entry on having special familiars is only found in the NPC section of the MM, PCs can't have special familiars--despite the fact that it actually gives the mechanism (the find familiar spell) by which said NPCs acquire their familiars. Such a DM would assert that it is the combination of the find familiar spell, and the magical blue glowing "NPC" halo that allows the summoning of special familiars.
Or a DM could be a more reasonable human being and assume that the text in the pseudodragon entry, combined with the specific variant trait listing for being a familiar, is supposed to be explaining the manner in which pseudodragons can be acquired, and take the p347 entry as providing additional information about it, namely that it requires the find familiar spell.
The crawling claw (or "some other Tiny monster") situation is a bit iffier. A reasonable angle for a DM to take could be either that you just use find familiar and convince them to become your familiar somehow (much the same as pseudodragon or imp), but it also isn't terribly unreasonable for a DM to say that it's going to be more involved to get such a non-standard familiar, and set some sort of role-playing requirements that may make it more trouble that it's worth for most players. In any case, I would assume that all special familiars share the ability to terminate their bond at any point.