Up to one of each. You cannot combine separate blast essences, or separate blast shapes.
A warlock can apply one eldritch essence invocation to a single eldritch blast, choosing from any of the eldritch essence invocations he knows.
(Complete Arcane pg. 8)
A warlock can apply only one blast shape at a time to an eldritch blast, and he can choose from any of the blast shape invocations that he knows.
(Complete Arcane pg. 9)
And the Eldritch Claws feat does not, as you might expect it to, give you a new blast shape invocation. It uses its own rules. Rules which say
You cannot use your normal eldritch blast ability while your eldritch claws exist.
(Dragon vol. 358 pg. 87)
Since you aren’t using eldritch blast itself, and the rules for Eldritch Claws don’t say anything about blast shapes or eldritch essences, you do not have any ability to apply any of those things to Eldritch Claws. Eldritch Claws are a separate ability that just happens to scale along with your eldritch blast.
And it’s not like the authors didn’t know how to word their way around this: cf. Grappling Blast on the same page, which specifies that you “discharge your eldritch blast against the foe you’re grappling,” and “You may not add any blast shape invocations to your eldritch blast when using it in this manner,” which means you can use eldritch essences (since you are actually using eldritch blast and not just using a separate ability that scaled along with eldritch blast).
For the record, the best options for your blast shape and eldritch essence—by far—are eldritch glaive and vitriolic blast. If you aren’t doing the glaive-lock thing, none of the other shapes are really worth the invocation slot to learn. Maybe eldritch chain; you’re still making touch attacks and you don’t do any worse against the primary target. But it’s a very odd campaign where the immense length of eldritch spear comes into play, and all the others give up your ability to make a touch attack, instead forcing saving throws, or requiring a real attack. Either vastly diminishes the likelihood of the eldritch blast getting through. The non-vitriolic essences have similar problems, as they often change eldritch blast into some resistible damage type (instead of irresistible untyped damage), and then then tack on some rider effects with low reliability; with the warlock’s extremely limited invocations known, is very hard to justify. The only reason vitriolic blast is good is because acid resistance isn’t all that common, and turning eldritch blast into SR: No gives you an option for dealing with the one thing the regular eldritch blast has difficulty with.