I provide a summary below of what is currently known (to my knowledge) regarding obfuscation of various class of "mostly-zero" functions.
From Indistinguishability Obfuscation
What we can obfuscate: all efficient functions
Which notion of security: indistinguishability obfuscation
Underlying assumptions: new, exotic, and relatively poorly understood assumptions.
Is it efficient? Nope.
You state in your question:
I also know that there is no generic scheme that can obfuscate all functions -- but perhaps it's possible to find a generic scheme that obfuscates all functions in this class.
However, as far as we know, it is perfectly possible that all efficiently computable functions (hence in particular all functions you are interested about) can be obfuscated with an iO scheme. We have a bunch of concrete candidates, and there has been a large body of work on the underlying assumptions. Providing a complete treatment would be out of scope for this question (I count more than 160 papers on the subject since 2013), but roughly, all known constructions of iO rely on exotic assumptions: either the existence of graded encoding schemes, whose security is poorly understood and with many (many) existing attacks, the latest being this one, or recent assumptions regarding the weak pseudorandomness of some extreme families of low depth pseudorandom generators -- some assumptions made in the three latest results (1, 2, 3) in this area have already been broken. Furthermore, none of the iO schemes we currently know of would be concretely efficient. Still, as a theoretical feasibility result, all functions you care about can be obfuscated (under exotic assumptions), where the obfuscation security notion is that of indistinguishability obfuscation.
From Multilinear Maps
What we can obfuscate: evasive low-degree polynomial modulo a prime
Which notion of security: virtual black box obfuscation (the strongest possible notion)
Underlying assumptions: new, exotic, and relatively poorly understood assumptions.
Is it efficient? Nope.
Barak et al. study in this paper the possibility of achieving even stronger notions of obfuscation (such as virtual grey box obfuscation, or virtual black box obfuscation) for the class of evasive functions. A class of function is evasive if, for any fixed input $x$, a random function from the class evaluates to $0$ on $x$ with overwhelming probability. While their results are mainly negative, Barak et al. also provide an interesting positive result: under a new perfectly-hiding multilinear encoding assumption, which is the same kind of assumption that is used in constructions of iO schemes (hence suffers from comparable weaknesses and gives comparable (in)efficiency), there exists a virtual black box obfuscator for evasive functions that tests if a low-degree polynomial evaluates to zero modulo some large prime. Virtual black box obfuscation is the ultimate security notion for obfuscation: while iO states that it should be infeasible to distinguish two functionally equivalent obfuscated circuits, VBB states that everything that the adversary sees, given the VBB obfuscation of the function, can be simulated given only black-box access to the function. It is known that VBB is in general impossible to achieve; hence, this work shows that this impossibility does not seem to hold anymore for this restricted class of evasive functions.
From LWE
What we can obfuscate: conjunctions, compute-and-compare functions
Which notion of security: distributional virtual black-box
Underlying assumptions: the well-studied (conjectured quantum-resistant) LWE assumption (or entropic variants of it)
Is it efficient? Not really, but depending on the primitive to obfuscate, it can be doable.
There has been recently a large body of work identifying interesting classes of "mostly-zero" functions, which can be obfuscated under the plain (or variants of) LWE assumption, which is a well-studied assumption.
1) Conjunctions. Several papers have proposed schemes to obfuscate functions of the form
$f : (x_1, \cdots, x_n) \rightarrow \bigwedge_{i \in I} y_i$,
where each $y_i$ is either $x_i$ or $1-x_i$, and $I$ is a subset of $[1, \cdots, n]$. This paper shows how to obfuscate this class of function under an "entropic variant" of the ring-LWE assumption. Obfuscating a 64-bit function takes a few hours, and evaluating the obfuscated program takes a few seconds. The security notion achieved is that of distributional VBB obfuscation, a variant of the notion of VBB obfuscation which also relies on the entropy of the obfuscated function.
2) Compute-and-compare functions. This is a very nice result achieved independently in this paper and in this paper. Assuming only the plain LWE assumption, they show how to compute a very large class of interesting evasive functions: the functions that have two secret values $(\alpha, \beta)$ hardcoded and, on input $x$, evaluate an arbitrary function $f$ on $x$ and output $\alpha$ if and only if $f(x) = \beta$ (and $\bot$ otherwise). This captures as a particular case conjunctions, point functions, but also much more. Their construction also achieves distributional virtual black-box obfuscation.
From Group-Based Assumptions
What we can obfuscate: conjunctions, pattern matching, hyperplanes membership
Which notion of security: distributional VBB
Underlying assumptions: from entropic DDH to the generic group model
Is it efficient? It can be reasonably efficient
Many constructions have also been designed under various assumptions in standard discrete-log-hard abelian groups. Most notably:
1) Conjunctions can also be obfuscated in the generic group model, and achieve distributional VBB, as shown by this very nice recent paper.
2) Pattern Matching. Two recent works have considered the task of obfuscating functions that test whether a given input string contains a pattern, with "wildcards" (i.e., the pattern can contain wildcards * that correspond to "any value there"), and achieve distributional VBB for patterns having some entropy. These are very nice results, and are in addition relatively efficient compared to everything that I've discussed so far. The first of these paper achieves security in the generic group model, while the second achieves security in the standard model, under a knowledge-of-exponent style assumption (i.e., something plausible, but far from being the most well understood assumptions).
3) Hyperplane Membership Testing. Eventually, this paper shows how to obfuscate hyperplane membership testing (i.e., checking whether an input vector belongs to a secret obfuscated hyperplane), which generalize point functions to a much larger class. The obfuscation is relatively efficient and is proven secure under a strong variant of the DDH assumption.
Note: there exists also DDH-based obfuscation scheme for re-encryption functions, but those do not fit into your notion of mostly-zero functions.
From One-Way Functions
What we can obfuscate: point functions
Which notion of security: distributional VBB
Underlying assumptions: exponentially strong one-way functions, one-way permutations, or deterministic encryption
Is it efficient? Yes
To conclude this overview, the weakest of all these "mostly-zero" functions that you want to obfuscate are point functions, which are equal to zero everywhere, except on a specific point. Such functions can be obfuscated, depending on the exact notion of obfuscation achieved, assuming the existence of exponentially strong one-way functions (here) or one-way permutation (here), or deterministic encryption (here) all being highly plausible assumptions (the second one being even a fundamental assumption of cryptography). Obfuscation of point functions has been studied in depth and has many applications in cryptography. Note that to achieve stronger notions of obfuscation for point functions (e.g. to achieve composability, or non-malleability), one typically rely on stronger DDH-like assumptions (e.g. here and here).
