Below is a sketch of what is and is not known today concerning Verdier’s two desiderata in positive characteristic.

1. “A wild Grothendieck‐group of constructible sheaves with a functorial push‐forward, which on stalks remembers both the rank and the wild ramification.”

–  **Still open in general.**  The naive push‐forward of constructible functions fails (Artin–Schreier shows exactly why), and nobody has exhibited a single “larger” abelian group  

   G₀^wild(X)  
of sheaves (or “functions + ramification data”) on which  
   f_! : G₀^wild(X) → G₀^wild(Y)  
makes sense and factors the usual Rf₁ (or Euler‐characteristic) push‐down.   
–  **Partial results:**  
  •  For *curves* one can adjoin the Swan conductors at points, à la Deligne–Laumon, and recover a functorial ε‐factor formalism; but in higher dimension—no complete theory.  
  •  Kato and Abbes–Saito have built a very fine ramification theory in higher dimensions, defining refined Swan conductors and characteristic classes of ramification, but so far only at the level of cohomological invariants, not as a covariant group of “wild‐decorated” sheaves.  

2. “A MacPherson‐style Chern‐class transformation in characteristic p.”

Here the story is more optimistic, *once one is willing to work with ℓ-adic sheaves (ℓ≠p), smooth ambient varieties, and invert ℓ*.  

A)  **Singular support and characteristic cycles**  
------------------------------------------------  
Over a perfect field of characteristic p, Beilinson (in lectures) and—more systematically—Takeshi Saito (in a sequence of preprints) have shown that if  
  X  is smooth over k,  
  F  is an ℓ-adic constructible complex of finite tor‐dimension,  
then there is a well‐defined *singular support*  
  SS(F)  ⊂  T*X  
a conical Lagrangian; and in fact a *characteristic cycle*  
  CC(F)  ∈  Z_d(SS(F))  
of the same dimension as X.  Moreover one has an index formula  
  χ_c(X, F)  =  ⟨CC(F), [0_X]⟩  
and, more generally, a functorial direct‐image theorem for CC under proper maps.  

B)  **From CC to Chern classes**  
--------------------------------  
Exactly as in the complex‐analytic situation one may now set  
  c(F)  :=  c(TX)  ∩ 0^! CC(F)   
where 0^! : CH_*(T*X) → CH_*(X)  is the Gysin map along the zero‐section.  One checks (again just as in MacPherson’s original argument) that  
  c(Rf_! F)  =  f_* c(F)  
in  CH_*(Y)[1/ℓ]  (or in ℓ-adic homology).  In particular, if one takes F to be the constant sheaf on a closed subvariety Z ⊂ X one recovers a cycle which in characteristic 0 is exactly the Chern–Schwartz–MacPherson class of Z.  Thus—*under the hypotheses*  

 •  X smooth,  
 •  coefficients in ℚ_ℓ  (ℓ≠p),  
 •  after inverting ℓ in your cycle groups,  

one *does* now have a MacPherson‐style transformation  
  K₀(D^b_c(X;ℚ_ℓ))  ––c(–)––>  CH_*(X)[1/ℓ]  
which extends to the usual constructible‐function formalism once one identifies constructible functions with classes of constant‐sheaf complexes.  

C)  **Euler obstruction and vanishing cycles**  
-----------------------------------------------  
The classical “local Euler obstruction” of a variety V ⊂ X at v is recovered in this ℓ-adic/microlocal picture as the *generic multiplicity* m_V(1_V) of CC(1_V) along the conormal T*_V X.  One then checks by the usual microlocal argument that for a general linear form ℓ: X→A¹ the vanishing‐cycles functor Rψ_ℓ kills all of those conormals not meeting dℓ, hence  
  φ_ℓ(m_V(1_V))  =  0  
exactly as in the complex case.  

3. What one still lacks
– A truly intrinsic “wild” Grothendieck group on which one can push‐forward Euler data without passing to ℓ-adic sheaves and inverting ℓ.
– A fully algebraic MacPherson class c_{SM}(–) defined for arbitrary (possibly singular) X over a field of char p without assuming X sits in a smooth ambient or requiring ℓ≠p and resolution of singularities.
– A “resolution‐free” Nash‐blowup/obstruction‐theoretic construction of Euler obstruction in char p.

In sum: Verdier’s grand vision is now realized in the ℓ-adic, microlocal world (thanks to the work of Abbes–Saito, Beilinson, Laumon, Kato, Saito …), subject to the usual ℓ≠p and smoothness hypotheses and inverting ℓ, but no purely algebraic, wild‐ramification analogue of the naïve constructible‐function/Euler‐characteristic formalism is known without those extra hypotheses.