  Criticality of chiral phase transition at finite temperature is investigated
in a soft-wall AdS/QCD model with $SU_L(N_f)\times SU_R(N_f)$ symmetry,
especially for $N_f=2,3$ and $N_f=2+1$. It is shown that in quark mass
plane($m_{u/d}-m_s$) chiral phase transition is second order at a certain
critical line, by which the whole plane is divided into first order and
crossover regions. The critical exponents $\beta$ and $\delta$, describing
critical behavior of chiral condensate along temperature axis and light quark
mass axis, are extracted both numerically and analytically. The model gives the
critical exponents of the values $\beta=\frac{1}{2}, \delta=3$ and
$\beta=\frac{1}{3}, \delta=3$ for $N_f=2$ and $N_f=3$ respectively. For
$N_f=2+1$, in small strange quark mass($m_s$) region, the phase transitions for
strange quark and $u/d$ quarks are strongly coupled, and the critical exponents
are $\beta=\frac{1}{3},\delta=3$; when $m_s$ is larger than
$m_{s,t}=0.290\rm{GeV}$, the dynamics of light flavors($u,d$) and strange
quarks decoupled and the critical exponents for $\bar{u}u$ and $\bar{d}d$
becomes $\beta=\frac{1}{2},\delta=3$, exactly the same as $N_f=2$ result and
the mean field result of 3D Ising model; between the two segments, there is a
tri-critical point at $m_{s,t}=0.290\rm{GeV}$, at which
$\beta=0.250,\delta=4.975$. In some sense, the current results is still at mean
field level, and we also showed the possibility to go beyond mean field
approximation by including the higher power of scalar potential and the
temperature dependence of dilaton field, which might be reasonable in a full
back-reaction model. The current study might also provide reasonable
constraints on constructing a realistic holographic QCD model, which could
describe both chiral dynamics and glue-dynamics correctly.
