plain strings (207)	foo
anchors (208)		k$
ranges (202)		^[a-f]*$
backrefs (201)		(...).*\1
abba (190)		^((?!(.)(.)\3\2).)*$
a man, a plan (177)	^(.)[^p].*\1$
prime (286)		^(?!(xx+)\1+$)
four (199)		(.)(.\1){3}
order (199)		^.{5}[^e]?$
triples (574)		^(([147]4|40|3[269]|9[05]|[378]1).+|0[369]*|[81][257])*$
glob (384)		(rr|ll|[lbr]o|en|ta|y|cr|eat|up).*\1
balance (251)		^((<>)*|<(?!>).*>)$    //five false positives.
powers (59)		^(.|(?!.{48}$)(..)*)$  //three false positives
longcount (218)		0..1..1.. (..0){3}..1* ..0..1..1..0.. ..11(.{9}1){2}
longcount2 (218)	0..1..1.. (..0){3}..1* ..0..1..1..0.. ..11(.{9}1){2}
alphabetical (180)	^(a[er]\w+ ?)*(ass\w+ ?)*(ast+ ?)*(e\w+ ?)*(n\w+ ?)*(r\w+ ?)*(s\w+ ?)*(t\w+ ?)*$ // 4 errors

These are regex.alf.nu solutions by Bisqwit. Usernames indicated are from Reddit unless otherwise noted.

Alternatives:

ranges (202)		[a-f]{4}               //made of abuse.
order (199)		^[^o]?.{5}$            //another way of writing the same thing.
powers (59)		^(x|(xx){1,4}|((((((x{16})\8?)\7?)\6?)\5?)\4?)\3?)$ //zero false positives,
           		                                                    //with thanks to Hrafnahnef.
alphabetical (156)	^(a[er]\w+ ?)*(asse[rn]\w+ ?)*(asse[st]\w+ ?)*(ast\w+ ?)*(e[an]\w+ ?)*(e[rts]\w+ ?)*(n\w+ ?)*((?:ra|re[anr])\w+ ?)*(rese\w+ ?)*(rest\w+ ?)*(ret\w+ ?)*(se\w+ ?)*(s[nt]\w+ ?)*(t\w+ ?)*$ // no false positives

Better solutions by others:

abba (193)		^(?!.*(.)(.)\2\1)                  // by Laugarhraun.
triples (596)		00($|3|6|9|12|15)|4.2|.1.+4|55|.17 // by alexandrosm @ [1]
glob (397)		ai|c$|^p|[bcnrw][bnopr] // by nwellnhof @ [2]
balance (287)		^(<(<(<(<(<(<.*)*>)*>)*>)*>)*>)*$  // by jensweh.
balance (288)		^(<(<(<(..)*>)*>)*>)*$ // by romanandreev @ [1]; some false positives
powers (80)		^(((x|x{8}|x{128})\3?)\2?)\1?$     // by pondscum; no false positives.
powers (97)		^(?!(.(..)+)\1*$) // by plby @ [1]; doesn't "cheat"
longcount (253)		^((.+)0 \2+1 ?)*$ // possibly by bbarry @ [3]
longcount2 (253)	^((.+)0 \2+1 ?)*$ // possibly by bbarry @ [3]
alphabetical (303)	r sn|( t\w+)\1|(tat|r). r|a t| ae|e e // by alexandrosm @ [1]

Best solutions that work in the spirit of the test without abusing the test-cases:
(Where different from what is posted above; recursion shortcomings are ignored)

a man, a plan (176)	^(.)(.).*\2\1$
triples (523)		^([0369]|[258][0369]*[147]|([258][0369]*[258]|[147])([0369]|[147][0369]*[258])*([258]|[147][0369]*[147]))*$                                                          // by Bisqwit.
balance (286)		^(<(<(<(<(<(<.*>)*>)*>)*>)*>)*>)*$                     // by Overv.
order (156)		^a*b*c*d*e*f*g*h*i*j*k*l*m*n*o*p*q*r*s*t*u*v*w*x*y*z*$ // by Athox.
glob (323)		^(?:(.+) .+ \1|(.*)\*(.*) .+ \2.+\3|(.*)\*(.*)\*(.*) .+ \4.+\5.+\6|\*(.*)\*(.*)\* .+ .+\7.+\8.+)$                                                                         // by Lozzer2
glob (333)		^(\*?)(\w*)(\*?)(\w*)(\*?)(\w*) .* ((.(?!\1))+|\1)\2((.(?!\3))+|\3)\4((.(?!\5))+|\5)\6$ // by hadrel @ [2]

[1]: https://gist.github.com/jonathanmorley/8058871
[2]: https://news.ycombinator.com/item?id=6941231
[3]: https://gist.github.com/jpsim/8057500