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I am not going to add any new examples but suggest a systematic way of looking at examples. If one looks at special phenomena in characteristic $2$ one can classify them as follows (though this division is far from clear cut):

1. They are really special to positive characteristic and not only characteristic $2$.
2. They are still really really positive characteristic phenomena but they only appear for some numerical invariants that depend on $p$ (normally increasing with $p$) and as $2$ is the smallest prime they appear "earlier" in characteristic $2$ and hence are encountered there first.
3. They are really special to characteristic $2$.

Some examples and their classification:

1. Here one can look at failure of strong versions of Bertini, for instance a base point free linear system all of whose members are singular (take the $p$'th power of a very ample linear system). This is uniform in $p$ (though if one starts with a characteristic free ample system, the degree will grow as $p$ grows so in that sense it could also be classified under 2).
2. The existence of quasi-elliptic fibrations in characteristic $2$ (and $3$) is an example as the same phenomemena of a regular but non-smooth curve over a non-perfect field exists in all positive characteristics. However, by a result of Tate the genus of such an example is bounded from below by a linear function of $p$ so they appear later and later. However, there is one further complication in that the quasi-elliptic case is of Kodaira dimension $0$ which makes $2$ and $3$ special as all other examples are of general type. This gives an example of overlap between 2) and 3).

Another such example is that of Enriques surfaces. On the one hand the Godeaux construction gives examples of smooth surfaces whose fundamental group scheme is any group scheme of order $p$ with various numerical invariants depending on $p$. However, only in characteristic $2$ (and $3$ I think) is it of Kodaira dimension $0$.

3. Here the examples that come to my mind are mostly somehow related to quadratic forms. They themselves of course really behave differently in characteristic $2$ (even purely geometrically) as does the orthogonal group. However, their influence goes further, for instance that theta characteristics behave differently in characteristic $2$ can be traced back to quadratic forms.

Addendum: To test my claim I went through the answers given so far and tried to classify as per above. Most of them are already mentioned above but two are not. First there is Jeremy's comment on torsion in $1+p\mathbb Z_p$ which on the face of it belongs to category 3). However, it is clearly related to $p$-adic radius of convergence of the logarithm and exponential series and that radius grows as $p$ grows. Hence, for absolutely ramified rings you can get the same phenomenon in all characteristics, what is special with $2$ is that it happens in the absolutely unramified. Note also that though the consequence mentioned by Jeremy is more arithmetic than algebro-geometric there are consequences of the latter type. Certainly, mixed characteristic ones such as the structure of finite group schemes but also for crystalline issues (technically the divided power structure on $2\mathbb Z_2$ is not nilpotent).

Sándor's example of failure of Kodaira vanishing is mostly of type 1) but the examples usually have $p$ as a parameter in numerical characters making it partly of type 2). There is even the fact that there are a few minimal surfaces of general type in characteristic $2$ (but in no other characteristic) for which $H^1(X,\omega_X^{-1})\ne 0$ which technically is of type 3).

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Addendum: To test my claim I went through the answers given so far and tried to classify as per above. Most of them are already mentioned above but two are not. First there is Jeremy's comment on torsion in $1+p\mathbb Z_p$ which on the face of it belongs to category 3). However, it is clearly related to $p$-adic radius of convergence of the logarithm and exponential series and that radius grows as $p$ grows. Hence, for absolutely ramified rings you can get the same phenomenon in all characteristics, what is special with $2$ is that it happens in the absolutely unramified. Note also that though the consequence mentioned by Jeremy is more arithmetic than algebro-geometric there are consequences of the latter type. Certainly, mixed characteristic ones such as the structure of finite group schemes but also for crystalline issues (technically the divided power structure on $2\mathbb Z_2$ is not nilpotent).

Sándor's example of failure of Kodaira vanishing is mostly of type 1) but the examples usually have $p$ as a parameter in numerical characters making it partly of type. There is even the fact that there are a few minimal surfaces of general type in characteristic $2$ (but in no other characteristic) for which $H^1(X,\omega_X^{-1})\ne 0$ which technically is of type 3).

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I am not going to add any new examples but suggest a systematic way of looking at examples. If one looks at special phenomena in characteristic $2$ one can classify them as follows (though this division is far from clear cut):

1. They are really special to positive characteristic and not only characteristic $2$.
2. They are still really really positive characteristic phenomena but they only appear for some numerical invariants that depend on $p$ (normally increasing with $p$) and as $2$ is the smallest prime they appear "earlier" in characteristic $2$ and hence are encountered there first.
3. They are really special to characteristic $2$.

Some examples and their classification:

1. Here one can look at failure of strong versions of Bertini, for instance a base point free linear system all of whose members are singular (take the $p$'th power of a very ample linear system). This is uniform in $p$ (though if one starts with a characteristic free ample system, the degree will grow with as $p$ grows so in that sense it could also be classified under 2).
2. The existence of quasi-elliptic fibrations in characteristic $2$ (and $3$) is an example as the same phenomemena of a regular but non-smooth curve over a non-perfect field exists in all positive characteristics. However, by a result of Tate the genus of such an example is bounded from below by a linear function of $p$ so they appear later and later. However, there is one further complication in that the quasi-elliptic case is of Kodaira dimension $0$ which makes $2$ and $3$ special as all other examples are of general type. This gives an example of overlap between 2) and 3). Another such example is that of Enriques surfaces. On the one hand the Godeaux construction gives examples of smooth surfaces whose fundamental group scheme is any group scheme of order $p$ with various numerical invariants depending on $p$. However, only in characteristic $2$ (and $3$ I think) is it of Kodaira dimension $0$.
3. Here the examples that come to my mind are mostly somehow related to quadratic forms. They themselves of course really behave differently in characteristic $2$ (even purely geometrically) as does the orthogonal group. However, their influence goes further, for instance that theta characteristics behave differently in characteristic $2$ can be traced back to quadratic forms.
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