Significantly more sightings over the years? This is probably something very easy for many of you, but I am not sure if I am doing this right. I am a biologist and have been looking at the occurrence (per year) of a certain species in a certain area from 2001 to 2016. As this is a rare species, in two years we saw none, in the other years we saw 1-10 individuals. It seems like the animals are being seen more in recent years and I think there is a significant increase of them over the years. But how do I test this? I have simplified my dataset to just years and a number of sightings in that year (of different individuals, so no double counts):

If I run a linear regression in SPSS my p-value is significant. But is this the right way to do it? 

 A: Small counts like with a few sightings of a rare animal are usually well modelled by a Poisson distribution, so Poisson regression is indicated.  Variability in number of sightings of a rare animal could well be caused by a varying level of effort, below I assume this is not the case, that is, effort is approximately constant over time. (If you have effort data, it could be included as an offset via the argument  offset=log(effort) in the glm command below).
I show below an analysis of your data using Poisson regression in R, which I think is more appropriate than the analysis you presents in the question. R code is:
year      <-  2001:2016
sighting  <-  c(1,0,3,2,0,4,2,2,5,4,4,10,8,3,6,5)
dat  <-  data.frame(year,sighting)
# Poisson regression model:
mod.0  <-  glm(sighting ~ year, family=poisson, data=dat)
summary(mod.0)
confint(mod.0)
plot(mod.0)

Which produces the following output: 
> mod.0  <-  glm(sighting ~ year, family=poisson, data=dat)
> summary(mod.0)

Call:
glm(formula = sighting ~ year, family = poisson, data = dat)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.0802  -0.7293  -0.2194   0.8220   2.0665  

Coefficients:
              Estimate Std. Error z value Pr(>|z|)    
(Intercept) -227.69850   61.49336  -3.703 0.000213 ***
year           0.11395    0.03058   3.726 0.000194 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 33.801  on 15  degrees of freedom
Residual deviance: 18.770  on 14  degrees of freedom
AIC: 67.106

Number of Fisher Scoring iterations: 5

> confint(mod.0)
Waiting for profiling to be done...
                    2.5 %       97.5 %
(Intercept) -351.90183933 -109.9909922
year           0.05539646    0.1757022

(I do not show here the plots, which do not indicate any problems with the model)
Note that the confidence interval on the slope parameter is far away from zero, so some increase in abundance with time is indicated.  With so few data effective model criticism (based on residuals and such) is not really possible, so you need to believe in the model!
