Remove legend ggplot 2.2

from r cookbook, where bp is your ggplot:

Remove legend for a particular aesthetic (fill):

bp + guides(fill="none")

It can also be done when specifying the scale:

bp + scale_fill_discrete(guide="none")

This removes all legends:

bp + theme(legend.position="none")

There might be another solution to this:
Your code was:

geom_point(aes(..., show.legend = FALSE))

You can specify the show.legend parameter after the aes call:

geom_point(aes(...), show.legend = FALSE)

then the corresponding legend should disappear

As the question and user3490026's answer are a top search hit, I have made a reproducible example and a brief illustration of the suggestions made so far, together with a solution that explicitly addresses the OP's question.

One of the things that ggplot2 does and which can be confusing is that it automatically blends certain legends when they are associated with the same variable. For instance, factor(gear) appears twice, once for linetype and once for fill, resulting in a combined legend. By contrast, gear has its own legend entry as it is not treated as the same as factor(gear). The solutions offered so far usually work well. But occasionally, you may need to override the guides. See my last example at the bottom.

# reproducible example:
library(ggplot2)
p <- ggplot(data = mtcars, aes(x = mpg, y = disp, group = gear)) +
geom_point(aes(color = vs)) +
geom_point(aes(shape = factor(cyl))) +
geom_line(aes(linetype = factor(gear))) +
geom_smooth(aes(fill = factor(gear), color = gear)) + 
theme_bw() 

Remove all legends: @user3490026

p + theme(legend.position = "none")

Remove all legends: @duhaime

p + guides(fill = FALSE, color = FALSE, linetype = FALSE, shape = FALSE)

Turn off legends: @Tjebo

ggplot(data = mtcars, aes(x = mpg, y = disp, group = gear)) +
geom_point(aes(color = vs), show.legend = FALSE) +
geom_point(aes(shape = factor(cyl)), show.legend = FALSE) +
geom_line(aes(linetype = factor(gear)), show.legend = FALSE) +
geom_smooth(aes(fill = factor(gear), color = gear), show.legend = FALSE) + 
theme_bw() 

Remove fill so that linetype becomes visible

p + guides(fill = FALSE)

Same as above via the scale_fill_ function:

p + scale_fill_discrete(guide = FALSE)

And now one possible answer to the OP's request

"to keep the legend of one layer (smooth) and remove the legend of the other (point)"

Turn some on some off ad-hoc post-hoc

p + guides(fill = guide_legend(override.aes = list(color = NA)), 
           color = FALSE, 
           shape = FALSE)  

If your chart uses both fill and color aesthetics, you can remove the legend with:

+ guides(fill=FALSE, color=FALSE)