Given the normal discrete logarithm problem:
$$a = b^c \mod{P}$$
with prime $P$ and numbers $a,b,c$
For which kind of $P,b$ the NFS/IC algorithm is faster than Baby-Step/Giant-Step+ Pollard's Rho ($\approx \mathcal{O}(\sqrt{q}) $)?
(with $q$ the biggest prime in factorization of $P-1$, with $P$ big prime)
Or in which cases NFS/IC it used?
Using its notation, the question is about the difficulty of the Discrete Logarithm Problem in a Schnorr Group modulo $P$, of prime order $q$. I'll assume $b^q\bmod P=1$ and $b\bmod P\ne1$.
That DLP problem is finding $c$ chosen at random in $[0,q)$ given $P$, $q$, $b$, and $a$ obtained as $b^c\bmod P$. Depending on parameters, the best known algorithms fall into two complexity classes:
somewhere between $\mathcal O(\sqrt{q}\,\ln P\,\ln\ln P)$ [in theory] and $\mathcal O(\sqrt{q}\,\ln^2 P)$ for Baby-Step/Giant-Step and it's practical improvement: Pollard's Rho with distinguished points (which can be efficiently distributed and requires little memory; see Paul C. van Oorschot and Michael J. Wiener, Parallel Collision Search with Cryptanalytic Applications, in Journal of Cryptology, 1999). The cost is often stated as $\mathcal O(\sqrt{q})$ multiplications of integers of size $P$, and this has been recently shown to cost $\mathcal O(\ln P\,\ln\ln P)$, see this.
something like $\exp\left( \left(\sqrt[3]{\frac{64}{9}} + o(1)\right)(\ln P)^{\frac{1}{3}}(\ln \ln P)^{\frac{2}{3}}\right)$, for the Number Field Sieve applied to the Discrete Logarithm (see this).
in which cases is NFS/Index Calculus used?
For a given size of $q$, the first class of algorithms (Pollard's Rho..) is best for large $P$. The second (NFS) is faster for relatively small $P$, including $q$ a Sophie Germain prime (equivalently, $P$ a safe prime).
For 256‑bit $q$, the first class of algorithm is better for 8192-bit $P$, the second for 512‑bit $P$. I prefer not digging where exactly the crossover is, or what's the exact difference between NFS and IC.
The exact cost of the number field sieve algorithm is somewhat fuzzy (the usually quoted complexity is only valid in a log-asymptotic sense). Lenstra and Verheul tried to capture a more usable version of the complexity which has been broadly accepted. For parameter sets of interest the related estimates published by NIST would probably be generally agreed on:
80-bits work: 160-bit $q$, 1024-bit $P$
112-bits work: 224-bit $q$, 2048-bit $P$
128-bits work: 256-bit $q$, 3072-bit $P$
192-bits work: 384-bit $q$, 7680-bit $P$
256-bits work: 512-bit $q$, 15360-bit $P$
