example of quintics with 5 ordinary triple point I know  we can bound the triple point on quintics in cp^3 by 5. But how to write down  quintics with 5 ordinary triple point (here are simple elliptic singularity)explicitly? 
 A: Choose five points $P_i$ in general linear position (all choices are
equivalent under ${\rm PGL}_4$, so you might as well put four of them
at the coordinate vectors and the fifth at $(1:1:1:1)$); then
at each $P_i$ the condition of a triple point imposes $1+3+6 = 10$
linear conditions on the space of quintics, which has dimension
${8 \choose 5} = 56$, so you get a linear system of quintics 
with a triple point at each $P_i$, of projective dimension
$56 - 5\cdot 10 - 1 = 5$.
A quick Google search  turns up a paper

Stephan Endrass, Ulf Persson, and Jan Stevens: 
  Surfaces with triple points,
  J. Algebraic Geom. 12 (2003), 307--320.
  arXiv: math/0010163

which gives on pages 5-6 an alternative description.  Start with a
cubic surface that has a triple point (so is a cone over a cubic 
plane curve), and apply a "reciprocal transformation": choose four points
$p_i$ on the cubic (other than the cubic singularity), use projective coordinates $(x_1:x_2:x_3:x_4)$ that make $p_i$ 
the coordinate points, and apply the birational involution
$$
(x_1:x_2:x_3:x_4) \leftarrow - \ - \rightarrow (1/x_1:1/x_2:1/x_3:1/x_4).
$$
As remarked at the end of this section, "The above analysis is very elementary, 
and parts of it have not too surprisingly already appeared in the literature",
citing a 1952 paper in Italian:

D. Gallarati, Sulle superficie del quinto ordine dotate di punti tripli,
  Rend. Accad. Naz. Lincei, serie VIII, XII (1952), 70--75.

