DatePicker is producing weird issues in the WASM head. 2 different problems are listed and are possibly related in this issue report.
DatePicker<Page ...>
<DatePicker />
</Page>
<Page
x:Class="SampleApp.MainPage"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
xmlns:controls="using:Microsoft.Toolkit.Uwp.UI.Controls"
mc:Ignorable="d" DataContext="{Binding Source={StaticResource Composition}, Path=Root}">
<controls:MasterDetailsView>
<controls:MasterDetailsView.NoSelectionContentTemplate>
<DataTemplate>
<DatePicker />
</DataTemplate>
</controls:MasterDetailsView.NoSelectionContentTemplate>
</controls:MasterDetailsView>
</Page>
If you open the dev menu, you will see this error:

<PackageReference Include="Uno.UI" Version="2.1.0-dev.1402" />
<PackageReference Include="Uno.UI.RemoteControl" Version="2.1.0-dev.1402" Condition="'$(Configuration)'=='Debug'" />
<PackageReference Include="Uno.Wasm.Bootstrap" Version="1.1.0-dev.422" />
<DotNetCliToolReference Include="Uno.Wasm.Bootstrap.Cli" Version="1.0.10" />
<PackageReference Include="Reactive.Wasm" Version="1.2.2" />
<PackageReference Include="ReactiveUI.Uno" Version="11.2.1" />
This is caused by datepicker's control-template. #2707 partially addressed this by fixing the template.
However, the selector for datepicker will requires additional works, either by re-implementing it from scratch, or use a native wrapper for datepicker using <input type='date' />, or mix of both.
There's a few date control issues, notably https://github.com/unoplatform/uno/issues/365.
It seems the DatePicker is still broken

While it renders this, no interaction is possible.
Most helpful comment
This is caused by datepicker's control-template. #2707 partially addressed this by fixing the template.
However, the selector for datepicker will requires additional works, either by re-implementing it from scratch, or use a native wrapper for datepicker using
<input type='date' />, or mix of both.